DATA AS INFORMATION Data can be either qualitative or quantitative. Examples of Qualitative Data Favorite colors Favorite foods Sunny or rainy Qualitative data describes characteristics about data such as colors, shapes, or preferences. Examples of Quantitative Data Temperatures for a week Inches of snowfall for a week Heights of tomato plants Quantitative data can be measured using numbers. It answers how many or how much? Write yes if the data set is qualitative. Write no if it is not. 1. Favorite sports [yes] 2. Heights of trees [no] 3. Cloudy or sunny [yes] 4. Favorite radio station [yes] 5. Distance to the sun [no] Write yes if the data set is quantitative. Write no if it is not. 6. Favorite television show [no] 7. Heights of giraffes [yes] 8. Lengths of earthworms [yes] 9. Ages of students [yes] 10. Favorite car [no] Discovering Math, Statistics and Data Analysis, Data as Information...or, Getting to Know It 1
DATA AS INFORMATION Data can be either qualitative or quantitative. Examples of Qualitative Data Favorite colors Favorite foods Sunny or rainy Qualitative data describes characteristics about data such as colors, shapes, or preferences. Examples of Quantitative Data Temperatures for a week Inches of snowfall for a week Heights of tomato plants Quantitative data can be measured using numbers. It answers how many or how much? Tell whether each data set is qualitative or quantitative. 1. Favorite sports [qualitative] 2. Inches of rain for a month [quantitative] 3. Colors of flowers [qualitative] 4. Favorite type of music [qualitative] 5. Favorite books [qualitative] 6. Heights of students [quantitative] 7. Weights of panda bears [quantitative] 8. Ages of elephants [quantitative] 9. Favorite games [qualitative] 10. Arm lengths [quantitative] 11. Basketball scores of a team for a season [quantitative] 12. Class percent scores on a history test [quantitative] 13. Favorite animal [qualitative] 14. Math scores for a class [quantitative] Discovering Math, Statistics and Data Analysis, Data as Information...or, Getting to Know It 2
DATA AS INFORMATION Qualitative data describes characteristics about data such as colors, shapes, or preferences. Favorite colors and whether a day is sunny or cloudy are examples of qualitative data. Quantitative data can be measured. It answers how many or how much? Temperature, length, height, weight, speed, age, or number of things are examples of quantitative data. Tell whether each data set is qualitative or quantitative. 1. Shapes of street signs [qualitative] 2. Inches of precipitation for a month [quantitative] 3. Speeds at which animals run [quantitative] 4. Favorite sports [qualitative] 5. Favorite foods [qualitative] 6. Weights of watermelons [quantitative] 7. Lengths of hummingbirds [quantitative] 8. Number of students in each class [quantitative] 9. Favorite video games [qualitative] 10. Ages of students [quantitative] 11. Baseball scores of a team for a season [quantitative] 12. Class grades on a spelling test [quantitative] 13. Colors of tinted windows [qualitative] 14. Number of sports cards in individual collections [quantitative] 15. Distances run by a runner in one month [quantitative] Discovering Math, Statistics and Data Analysis, Data as Information...or, Getting to Know It 3
DATA DISTRIBUTION A line plot uses a number line to show how data are spread out. This line plot shows the ages of children in a karate club. The oldest child is 12 and 5 6 7 8 9 10 11 12 the youngest is 6. Ages To find the range of the ages, subtract: 12 6 6. The range of the ages is 6 years. The line plot shows that one child is 7 years old, none are 11 years old, and four are 8 years old. The line plot shows the heights, in inches, of children in a third grade class. Use data from the line plot to answer each question. 35 36 37 38 39 40 41 42 43 44 45 Heights (in inches) 1. How tall is the tallest child? [45 inches] 2. How tall is the shortest child? [37 inches] 3. What is the range of the heights? [ 8 inches] 4. At which height is the greatest number of children? [39 inches] 5. How many children are 44 inches tall? [none] 6. How many more children are 41 inches tall than 42 inches tall? CHALLENGE How many children are in the class? [21 students] Discovering Math, Statistics and Data Analysis, Data Distribution...or, Plotting Along 1
DATA DISTRIBUTION A line plot uses a number line to show how data are spread out. The line plot to the right shows the number of pets of students in a fourth grade class. To find the range of the number of pets, find the difference between the greatest number and the least number: 6 0 = 6. The range of the number of pets is 6. 0 1 2 3 4 5 6 Number of Pets Use the line plot to interpret the data. The greatest number of students has1 pet. No students have 5 pets and 1 student has 6 pets. In general, fewer students have greater numbers of pets. Use data from the line plot above to answer each question. 1. How many students are in the class? [23 students] 2. How many students have pets? [17 students] 3. How many students have 3 or more pets? [4 students] 4. How many more students have 2 pets than have 4 pets? 5. How many students have 1 or 2 pets? [13 students] 6. Do more students have no pets or 2 pets? How many more? [no pets; 1 more] 7. Suppose a new student joins the class who has 9 pet fish. How would the range of the data change? CHALLENGE Survey the students in your class. Make a line plot to show the number of pets students in your class have. [Check students work.] Discovering Math, Statistics and Data Analysis, Data Distribution...or, Plotting Along 2
DATA DISTRIBUTION A line plot uses a number line to show how data are spread out. The line plot to the right shows the number of CDs of students in a fifth grade class. To find the range of the number of CDs, find the difference between the greatest number and the least number: 15 5 10. The range of the number of CDs is 10. 5 6 7 8 9 10 11 12 13 14 15 Number of CDs Use the line plot to interpret the data. Notice that more students have between 5 and 9 CDs than have between 10 and 15 CDs. After 9 CDs, the number of CDs students have decreases as the number of CDs increases. Use data from the line plot above to answer each question. 1. What is the least number of CDs students in the class have? [5 CDs] 2. How many students have 12 CDs? [no students] 3. How many students have 9 CDs? [7 students] 4. How many more students have 7 CDs than have 6 CDs? 5. How many students have 8, 9, or 10 CDs? [11 students] 6. Do more students have 5 CDs or 10 CDs? How many more? 7. How many students have at least 8 CDs? [15 students] 8. How many students are in the class? [27 students] 9. Suppose a new student joins the class who has 18 CDs. How would the range of the data change? CHALLENGE Survey the students in your class. Make a line plot to show the number of CDs students in your class have. [Check students work.] Discovering Math, Statistics and Data Analysis, Data Distribution...or, Plotting Along 3
SUMMARIZING DATA The median, mode, and range are used to summarize data. At a diving practice, Maria received the following scores for her dive from a number of teammates: 6, 5, 7, 7, 6, 8, 6. Find the median, mode, and range of the scores. Put the scores in order from least to greatest: 5, 6, 6, 6, 7, 7, 8. The median is the middle number in the data set when the data are arranged in order: 5, 6, 6, 6, 7, 7, 8. middle number The mode is the number that occurs most often: 5, 6, 6, 6, 7, 7, 8. There may be more than one mode or no mode at all. To find the range, subtract the least number from the greatest number. 8 5 3 The median is 6, the mode is 6, and the range is 3. Each problem shows diving scores for different dives at the practice. Find the median, mode, and range of each data set. 1. Timothy s scores: 9, 6, 7, 9, 8 Median: [8] Mode: [9] Range: [3] 2. Jason s scores: 9, 6, 7, 9, 7, 5, 7 Median: [7] Mode: [7] Range: [4] 3. Jessica s scores: 5, 8, 6, 8, 7, 9, 6 Median: [7] Mode: [6 and 8] Range: [4] 4. Colin s scores: 5, 7, 6, 7, 5, 5, 7, 5, 6 Median: [6] Mode: [5] Range: [2] 5. Lucy s scores: 8, 10, 7, 9, 8, 7, 7, 8, 7 Median: [8] Mode: [7] Range: [3] Discovering Math, Statistics and Data Analysis, Summarizing Data...or, Around the Middle 1
SUMMARIZING DATA The median, mode, and range are used to summarize data. The median and the mode indicate central or typical data. The range shows the spread of the data. At a snowboarding practice, Josh received the following scores from a number of teammates: 6, 8, 9, 6, 6, 9, 5, 8 Find the median, mode, and range of the scores. Put the scores in order from least to greatest: 5, 6, 6, 6, 8, 8, 9, 9. The median is the middle number in the data set when the data are arranged in order: 5, 6, 6, 6, 8, 8, 9, 9 There are an even number of data points. middle numbers Add the two middle numbers. Divide the sum by 2 to find the median. 6 8 14 14 2 7 The mode is the number that occurs most often: 5, 6, 6, 6, 8, 8, 9, 9. There may be more than one mode or no mode at all. The median is 7, the mode is 6, and the range is 4. Find the median, mode, and range of each data set. 1. Julia s ice skating scores: 9, 8, 8, 7, 8, 6, 7, 8, 6 Median: [8] Mode: [8] Range: [3] 2. Calvin s skateboarding scores: 4, 8, 5, 7, 8, 8, 7, 5 To find the range, subtract the least number from the greatest number. 9 5 4 Median: [7] Mode: [8] Range: [4] 3. Class heights in inches: 53, 48, 46, 51, 49, 52 Median: [50] Mode: [none] Range: [7] Discovering Math, Statistics and Data Analysis, Summarizing Data...or, Around the Middle 2
SUMMARIZING DATA The mean, median, mode, and range are used to summarize data. The mean, the median, and the mode indicate central or typical data. The range shows the spread of the data. At a diving practice, Nadia received the following scores from a number of teammates: 9, 6, 8, 6, 6, 7 Find the mean, median, mode, and range of the scores. The mean is the average of the numbers in the data set. To find the mean, add the numbers in a data set. 9 6 8 6 6 7 42 Then divide the sum by the number of addends. 42 6 7 Put the scores in order from least to greatest: 6, 6, 6, 7, 8, 9. The median is the middle number in the data set. 6, 6, 6, 7, 8, 9 There are an even number of data points. Add the two middle numbers. Divide the sum by 2 to find the median. 6 7 13 13 2 6.5 The mode is the number that occurs most often. 6, 6, 6, 7, 8, 9. To find the range, subtract the least number from the greatest number. There may be more than one mode or no mode at all. 9 6 3 The mean is 7, the median is 6.5, the mode is 6, and the range is 3. Find the mean, median, mode, and range of each data set. 1. Melanie s snowboarding scores: 9, 8, 5, 7, 10, 9 Mean: [8] Median: [8.5] Mode: [9] Range: [5] 2. Calvin s diving scores: 6, 9, 5, 8, 6, 9, 7, 6, 7 Mean: [7] Median: [7] Mode: [6] Range: [4] 3. Math quiz scores: 84, 88, 94, 85, 82 Mean: [86.6] Median: [85] Mode: [none] Range: [12] Discovering Math, Statistics and Data Analysis, Summarizing Data...or, Around the Middle 3
GRAPHING DATA The table shows the favorite pets of a group of third graders. You can make a bar graph to display and interpret the data. Favorite Pets Pet Cat Dog Ferret Fish Bird Number of Students 8 10 3 5 2 To make a bar graph of the data in the table: 12 FAVORITE PETS Step 1 Step 2 Step 3 Step 4 Draw and label the side and bottom of the graph. Write the title at the top. Start with 0 and choose a scale. Here, multiples of 2 allow it to be clear and not too crowded. Write the numbers along one side. List the pets to label the bars. Draw bars to match the data. From the graph, you can see quickly that the most popular pet was dogs. Use the data in the table to make a bar graph. Number of Students 10 8 6 4 2 0 Cat Dog Ferret Fish Bird Pet Favorite Sport Sport Soccer Skateboard Cycling Basketball Softball Number of 8 4 8 6 3 Students Discovering Math, Statistics and Data Analysis, Graphing Data...or, Charting a Course 1
GRAPHING DATA The table shows the belt levels of students in a karate club. You can make a bar graph to display and interpret the data. Color Number of Students Belt Levels White Yellow Green Blue Brown Black 25 20 25 15 10 5 To make a bar graph of the data in the table: Step 1 Step 2 Step 3 Step 4 Draw and label each axis. Write the title at the top. Start with 0 and choose a scale. Here intervals of 5 allow it to be clear and not too crowded. Write the numbers along one axis. List the belt colors to label the bars, on the other axis. Draw bars to match the data. From the graph, you can see quickly which belts are held by more students and which by less. Use the data in the table to make a bar graph. Number of Students 30 25 20 15 10 5 0 BELT S White Yellow Green Blue Brown Black Belt Color Favorite Activity Activity Shop Movies Sports Video Read Games Number of Students 10 10 25 15 5 Discovering Math, Statistics and Data Analysis, Graphing Data...or, Charting a Course 2
GRAPHING DATA The table shows the amount of money raised at a School Book Fair from 2001 through 2005. You can make a line graph to show changes over time. To make a line graph of the data in the table: Step 1 Step 2 Step 3 Step 4 Step 5 Book Fair Sales Year 2001 2002 2003 2004 2005 Money Raised $200 $400 $450 $600 $650 Draw and label the axes. Title the graph. Start with 0 and choose a scale for the numerical axis. Here, intervals of 100 allow it to be clear and not too crowded. Write the numbers along the vertical axis. List the years along the horizontal axis. Plot the points on the graph. Connect the points in order. Money Raised (dollars) 700 600 500 400 300 200 100 0 BOOK FAIR SALES 2001 2002 2003 2004 2005 Year From the graph, you can see that the money raised has increased every year. Use the data in the table to make a line graph. Magazine Sales Year 2000 2001 2002 2003 2004 Number 75 125 100 150 175 Sold Discovering Math, Statistics and Data Analysis, Graphing Data...or, Charting a Course 3
READING GRAPHS Graphs are visual ways to show data. A bar graph uses bars to display the data. The bar graph shows the pets of third graders at Washington Elementary School. You can use the graph to interpret the data. The bar for dogs is the longest bar in the graph. It shows that more students have dogs than have any other type of pets. The bar for birds shows that 8 students have birds for pets. Number of Students 30 20 10 0 OUR PETS Dogs Cats Fish Birds Horses Other Type of Pet Use the bar graph to answer each question. 1. How many students have cats for pets? [18 students] 2. How many more students have dogs than cats for pets? [2 students] 3. Which two types of pets do the same number of students have? [fish and horses] 4. Natalie has a pet ferret. Which bar on the graph includes her pet? [the bar labeled other ] 5. How many students have birds or fish for pets? [13 students] 6. Do more students have cats or horses for pets? How do you know? [Cats; because the bar for cats is taller than the bar for horses.] 7. Do more students have birds or fish for pets? How do you know? [Birds; because the bar for birds is taller than the bar for fish.] CHALLENGE How many third graders at Washington Elementary have pets, if each student with pets has only one kind of pet? [66 students] Discovering Math, Statistics and Data Analysis, Reading Graphs...or, Story Lines 1
READING GRAPHS Graphs are visual ways to show data. A circle graph displays data on how parts of a set, a quantity, or a thing make up the whole set, quantity, or thing. The circle graph shows how a class spends their 1 hour PE class. You can use the graph to interpret the data. The graph shows that about 1 4 of the time is spent building strength. One quarter of an hour is 15 minutes. So the class spends about 15 minutes building strength. OUR 1 HOUR PE CLASS Aerobics Build Strength Warm-up Cool down Use the circle graph to answer each question. 1. Which activity takes up the greatest part of the class? Explain. [Aerobics, because it is the greatest part of the graph.] 2. Which two activities take about the same amount of time? [warm-up and cool down] 3. Is more time spent building strength or cooling down? [building strength] 4. About how much time is spent doing aerobics? Explain. [About 30 minutes since about half the graph is for aerobics and half of 1 hour is 30 minutes.] 5. About how much time is spent doing aerobics and building strength? Explain. [About 45 minutes since 30 min + 15 min = 45 min] CHALLENGE Is it reasonable to conclude that about 15 minutes of class time are spent on the warm-up from the graph? Explain. [No. Possible answer: Together the warm-up and cool down make up about 1 4 of the graph and one quarter of an hour is 15 minutes. So the time must be less than 15 minutes.] Discovering Math, Statistics and Data Analysis, Reading Graphs...or, Story Lines 2
READING GRAPHS Graphs are visual ways to show data. A line graph can be used to show changes over time. The line graph shows the high temperature recorded each day for a 7-day period. You can use the graph to interpret the data. The graph shows that the high temperature on Monday was 78 F. It also shows that the high temperature increased from Monday to Tuesday. Temperature ( F) 90 88 86 84 82 80 78 76 74 72 70 0 HIGH TEMPERATURES Mon. Tues. Wed. Thurs. Fri. Sat. Sun. Day Use the line graph to answer each question. 1. What was the high temperature on Thursday? [86 F] 2. Was the high temperature greater on Tuesday or on Thursday? How do you know? [Thursday; because the point for Thursday is higher on the graph than the point for Tuesday.] 3. How did the high temperature change from Wednesday to Thursday? [it increased by 2 F] 4. How much warmer was the high temperature on Friday than on Monday? [9 warmer on Friday] 5. Between which two days was there no change in high temperature? How does the graph show this? [Between Friday and Saturday. The line between the points for these days is horizontal.] 6. What pattern does the graph show in the high temperatures over the week? [the temperatures increased over the week] Discovering Math, Statistics and Data Analysis, Reading Graphs...or, Story Lines 3
COLLECTING AND HANDLING DATA Different types of graphs can be used to display different types of data. Line Graph A line graph shows changes of one variable with another, often over time. Use a line graph to show the weight gain of a puppy during its first year. Weight (pounds) 25 20 15 10 5 0 WEIGHT OF A PUPPY J F M A M J J A S O N D Month Bar Graph A bar graph displays data for categories or groups, for easy comparison. Use a bar graph to compare classmates favorite colors. Number of Students 12 10 8 6 4 2 0 FAVORITE COLORS Blue Red Green Yellow Color Circle Graph A circle graph displays the parts of a whole, for easy comparison. Use a circle graph to show how you spend each part of a typical day. School MY DAY Meals Sleep Sports Homework Which type of graph would you use to display each data set? Write line graph, bar graph, or circle graph. 1. Show the number of inches a baby grows during its first year. 2. Show what a child does with different portions of all the birthday money. [circle graph] 3. Show how many third-graders have different favorite breakfast foods at a school. [bar graph] 4. Show the amount of rainfall each month for 6 months. 5. Show number of students with different belts at a karate school. 6. Show how time is divided among all the activities of a science class. [circle graph] [line graph] [line graph or bar graph] [bar graph] 7. Show how the height of a plant changes during the year. [line graph] Discovering Math, Statistics and Data Analysis, Collecting and Handling Data...or, Form Counts 1
COLLECTING AND HANDLING DATA Different types of graphs can be used to display different types of data. Line Graph A line graph shows changes of one variable with another, often with time. Use a line graph to show the low temperature each day for 5 days. Bar Graph A bar graph displays data for categories or groups, for easy comparison. Use a bar graph to compare attendance at a book fair for 4 days. Circle Graph A circle graph displays the parts of a whole, for easy comparison. Use a circle graph to show how you spend your allowance each month. MY ALLOWANCE Entertainment Savings Food Games Which type of graph would you use to display each data set? Write line graph, bar graph, or circle graph. 1. Show the weight gain of a baby during its first year. [line graph] 2. Show how a child spends a typical day. [circle graph] 3. Show numbers of different types of rocks in a rock collection. [bar graph or circle graph] 4. Show the numbers of home runs of different baseball players. [bar graph] 5. Show the number of skateboards sold each year for 5 years. [bar graph or line graph] 6. Show how time is spent at a typical camp day. [circle graph] 7. Show how the height of an office building changes as it is being built. [line graph] 8. Show the different types of books in a library. [circle graph] Discovering Math, Statistics and Data Analysis, Collecting and Handling Data...or, Form Counts 2
COLLECTING AND HANDLING DATA Different types of graphs can be used to display different types of data. Double line and double bar graphs show two sets of data on one graph, making comparisons easy. Both types of graphs have keys that distinguish the sets of data. Double Line Graph A double line graph compares two different sets of data on one graph. Each line shows how the value of one variable changes with another, often with time. Double Bar Graph A double bar graph shows two different sets of data on one graph. It compares data by categories. Which type of graph would best display each data set? Write double line graph, double bar graph or either. 1. Show daily number of DVD rentals at two different stores over a 4-week period. [either] 2. Show attendance of boys and girls at a carnival from Thursday through Sunday. [either] 3. Show average lengths of males and females of different species of dolphins. [double bar graph] 4. Show temperature each hour for 2 different cities over an 8-hour period. [double line graph] 5. Show costs of two different brands of skateboards in 2000 and in 2005. [double bar graph] 6. Show monthly savings, (month by month) of a brother and sister for 6 months. [either] Discovering Math, Statistics and Data Analysis, Collecting and Handling Data...or, Form Counts 3
SAMPLE SIZE Data can be gathered using a survey. When a survey is conducted, the whole group being studied is called the population. A sample is the part of the population that participates in the survey. A sample is supposed to represent the population. A larger sample generally does so better than a smaller sample. Suppose a class wants to determine the favorite sport of students who attend their school. A survey of 50 students would better represent the school population than a sample of 5 students. A survey of 5 randomly chosen students from each class would better represent the school population than students on soccer teams. For each survey, tell which sample better represents the population. 1. Favorite grocery store of shoppers in a region. (a) A survey of 50 people currently shopping in a grocery store. (b) A survey of 50 people whose names are chosen from the telephone book. [b] 2. Favorite teacher of students in a school. (a) A survey of 20 students in the same class at a school. (b) A survey of 100 students from throughout the school. [b] 3. Favorite sport of students in a school. (a) A survey of 50 students in the school. (b) A survey of 50 teachers in the school. [a] 4. Ladybug species on a farm. (a) A collection of 10 ladybugs from one tree. (b) A collection of 30 ladybugs from around the farm. Discovering Math, Statistics and Data Analysis, Sample Size...or, The More the Merrier 1
SAMPLE SIZE A common method used to gather data is a survey. When a survey is conducted, the whole group being studied is called the population. A sample is the part of the population that participates in the survey. A sample is supposed to represent the population of interest. Because there may be differences between individuals in a population, a larger sample is generally better than a smaller sample. Suppose a student wants to determine the favorite sport of students at school. A survey of 50 students would better represent the school population than a sample of 5 students. A survey of 5 randomly chosen students from each class would better represent the school population than the members of the basketball team. For each survey, choose which sample better represents the population. Then tell why. [Possible answers are given for explanations.] 1. Favorite grocery store of shoppers in a region. (a) A survey of 50 people currently shopping in a grocery store. (b) A survey of 50 people whose names are chosen from the telephone book. [b; because many shoppers in the grocery store tend to shop there because they prefer that store] 2. Favorite teacher of students in a school. (a) A survey of 50 students in the school. (b) A survey of 50 teachers in the school. [a; because teachers are not in the population of interest] 3. The wingspan of monarch butterflies. (a) Measure the wingspan of 3 butterflies. (b) Measure the wingspan of 10 butterflies. [b; because a sample of 10 has a greater chance of representing the full range of wingspans than a sample of 3.] Discovering Math, Statistics and Data Analysis, Sample Size...or, The More the Merrier 2
SAMPLE SIZE A common method used to gather data is a survey. When a survey is conducted, the whole group being studied is called the population. Because not everyone in the population can be contacted, a sample is used. A sample is the part of the population that participates in the survey. A sample is supposed to represent the population of interest. Because there may be differences between individual members of a population, the larger a random sample is the more representative it is of the population. Suppose a scientist wants to measure the range of tree heights in a forest. A random sample of 50 tree heights would better represent the range of tree heights than a random sample of 5 tree heights. A random sample of 20 trees from throughout the forest would better represent the range of tree heights than a sample of 20 trees only from an area that has the tallest trees. For each survey, choose which sample better represents the population. Then tell why. [Possible answers are given for explanations.] 1. The wingspan of monarch butterflies. (a) Measure the wingspan of 5 butterflies. (b) Measure the wingspan of 20 butterflies. [b; because a sample of 20 is more likely to reflect the full range of variation of wingspan in the population than a smaller sample of 5.] 2. Favorite school subject of students in a school. (a) A survey of 50 students in the school. (b) A survey of 50 teachers in the school. [a; because teachers are not in the population of interest] 3. Species of trees in a valley. (a) Study 20 trees along the edge of the river. (b) Study 20 trees throughout the valley. [b; because different tree species may grow away from the river.] Discovering Math, Statistics and Data Analysis, Sample Size...or, The More the Merrier 3
DATA AS INFORMATION Data can be either qualitative or quantitative. Examples of Qualitative Data Favorite colors Favorite foods Sunny or rainy Qualitative data describes characteristics about data such as colors, shapes, or preferences. Examples of Quantitative Data Temperatures for a week Inches of snowfall for a week Heights of tomato plants Quantitative data can be measured using numbers. It answers how many or how much? Write yes if the data set is qualitative. Write no if it is not. 1. Favorite sports [yes] 2. Heights of trees [no] 3. Cloudy or sunny [yes] 4. Favorite radio station [yes] 5. Distance to the sun [no] Write yes if the data set is quantitative. Write no if it is not. 6. Favorite television show [no] 7. Heights of giraffes [yes] 8. Lengths of earthworms [yes] 9. Ages of students [yes] 10. Favorite car [no] Discovering Math, Statistics and Data Analysis, Data as Information...or, Getting to Know It 1
DATA AS INFORMATION Data can be either qualitative or quantitative. Examples of Qualitative Data Favorite colors Favorite foods Sunny or rainy Qualitative data describes characteristics about data such as colors, shapes, or preferences. Examples of Quantitative Data Temperatures for a week Inches of snowfall for a week Heights of tomato plants Quantitative data can be measured using numbers. It answers how many or how much? Tell whether each data set is qualitative or quantitative. 1. Favorite sports [qualitative] 2. Inches of rain for a month [quantitative] 3. Colors of flowers [qualitative] 4. Favorite type of music [qualitative] 5. Favorite books [qualitative] 6. Heights of students [quantitative] 7. Weights of panda bears [quantitative] 8. Ages of elephants [quantitative] 9. Favorite games [qualitative] 10. Arm lengths [quantitative] 11. Basketball scores of a team for a season [quantitative] 12. Class percent scores on a history test [quantitative] 13. Favorite animal [qualitative] 14. Math scores for a class [quantitative] Discovering Math, Statistics and Data Analysis, Data as Information...or, Getting to Know It 2
DATA AS INFORMATION Qualitative data describes characteristics about data such as colors, shapes, or preferences. Favorite colors and whether a day is sunny or cloudy are examples of qualitative data. Quantitative data can be measured. It answers how many or how much? Temperature, length, height, weight, speed, age, or number of things are examples of quantitative data. Tell whether each data set is qualitative or quantitative. 1. Shapes of street signs [qualitative] 2. Inches of precipitation for a month [quantitative] 3. Speeds at which animals run [quantitative] 4. Favorite sports [qualitative] 5. Favorite foods [qualitative] 6. Weights of watermelons [quantitative] 7. Lengths of hummingbirds [quantitative] 8. Number of students in each class [quantitative] 9. Favorite video games [qualitative] 10. Ages of students [quantitative] 11. Baseball scores of a team for a season [quantitative] 12. Class grades on a spelling test [quantitative] 13. Colors of tinted windows [qualitative] 14. Number of sports cards in individual collections [quantitative] 15. Distances run by a runner in one month [quantitative] Discovering Math, Statistics and Data Analysis, Data as Information...or, Getting to Know It 3
DATA DISTRIBUTION A line plot uses a number line to show how data are spread out. This line plot shows the ages of children in a karate club. The oldest child is 12 and 5 6 7 8 9 10 11 12 the youngest is 6. Ages To find the range of the ages, subtract: 12 6 6. The range of the ages is 6 years. The line plot shows that one child is 7 years old, none are 11 years old, and four are 8 years old. The line plot shows the heights, in inches, of children in a third grade class. Use data from the line plot to answer each question. 35 36 37 38 39 40 41 42 43 44 45 Heights (in inches) 1. How tall is the tallest child? [45 inches] 2. How tall is the shortest child? [37 inches] 3. What is the range of the heights? [ 8 inches] 4. At which height is the greatest number of children? [39 inches] 5. How many children are 44 inches tall? [none] 6. How many more children are 41 inches tall than 42 inches tall? [3 more children] CHALLENGE How many children are in the class? [21 students] Discovering Math, Statistics and Data Analysis, Data Distribution...or, Plotting Along 1
DATA DISTRIBUTION A line plot uses a number line to show how data are spread out. The line plot to the right shows the number of pets of students in a fourth grade class. To find the range of the number of pets, find the difference between the greatest number and the least number: 6 0 = 6. The range of the number of pets is 6. 0 1 2 3 4 5 6 Number of Pets Use the line plot to interpret the data. The greatest number of students has1 pet. No students have 5 pets and 1 student has 6 pets. In general, fewer students have greater numbers of pets. Use data from the line plot above to answer each question. 1. How many students are in the class? [23 students] 2. How many students have pets? [17 students] 3. How many students have 3 or more pets? [4 students] 4. How many more students have 2 pets than have 4 pets? [4 more students] 5. How many students have 1 or 2 pets? [13 students] 6. Do more students have no pets or 2 pets? How many more? [no pets; 1 more] 7. Suppose a new student joins the class who has 9 pet fish. How would the range of the data change? [The new range would be 9 0, or 9.] CHALLENGE Survey the students in your class. Make a line plot to show the number of pets students in your class have. [Check students work.] Discovering Math, Statistics and Data Analysis, Data Distribution...or, Plotting Along 2
DATA DISTRIBUTION A line plot uses a number line to show how data are spread out. The line plot to the right shows the number of CDs of students in a fifth grade class. To find the range of the number of CDs, find the difference between the greatest number and the least number: 15 5 10. The range of the number of CDs is 10. 5 6 7 8 9 10 11 12 13 14 15 Number of CDs Use the line plot to interpret the data. Notice that more students have between 5 and 9 CDs than have between 10 and 15 CDs. After 9 CDs, the number of CDs students have decreases as the number of CDs increases. Use data from the line plot above to answer each question. 1. What is the least number of CDs students in the class have? [5 CDs] 2. How many students have 12 CDs? [no students] 3. How many students have 9 CDs? [7 students] 4. How many more students have 7 CDs than have 6 CDs? 5. How many students have 8, 9, or 10 CDs? [11 students] 6. Do more students have 5 CDs or 10 CDs? How many more? 7. How many students have at least 8 CDs? [15 students] 8. How many students are in the class? [27 students] [2 students] 9. Suppose a new student joins the class who has 18 CDs. How would the range of the data change? [The new range would be 18 5, or 13.] CHALLENGE Survey the students in your class. Make a line plot to show the number of CDs students in your class have. [Check students work.] [5 CDs; 1 more] Discovering Math, Statistics and Data Analysis, Data Distribution...or, Plotting Along 3
SUMMARIZING DATA The median, mode, and range are used to summarize data. At a diving practice, Maria received the following scores for her dive from a number of teammates: 6, 5, 7, 7, 6, 8, 6. Find the median, mode, and range of the scores. Put the scores in order from least to greatest: 5, 6, 6, 6, 7, 7, 8. The median is the middle number in the data set when the data are arranged in order: 5, 6, 6, 6, 7, 7, 8. middle number The mode is the number that occurs most often: 5, 6, 6, 6, 7, 7, 8. There may be more than one mode or no mode at all. To find the range, subtract the least number from the greatest number. 8 5 3 The median is 6, the mode is 6, and the range is 3. Each problem shows diving scores for different dives at the practice. Find the median, mode, and range of each data set. 1. Timothy s scores: 9, 6, 7, 9, 8 Median: [8] Mode: [9] Range: [3] 2. Jason s scores: 9, 6, 7, 9, 7, 5, 7 Median: [7] Mode: [7] Range: [4] 3. Jessica s scores: 5, 8, 6, 8, 7, 9, 6 Median: [7] Mode: [6 and 8] Range: [4] 4. Colin s scores: 5, 7, 6, 7, 5, 5, 7, 5, 6 Median: [6] Mode: [5] Range: [2] 5. Lucy s scores: 8, 10, 7, 9, 8, 7, 7, 8, 7 Median: [8] Mode: [7] Range: [3] Discovering Math, Statistics and Data Analysis, Summarizing Data...or, Around the Middle 1
SUMMARIZING DATA The median, mode, and range are used to summarize data. The median and the mode indicate central or typical data. The range shows the spread of the data. At a snowboarding practice, Josh received the following scores from a number of teammates: 6, 8, 9, 6, 6, 9, 5, 8 Find the median, mode, and range of the scores. Put the scores in order from least to greatest: 5, 6, 6, 6, 8, 8, 9, 9. The median is the middle number in the data set when the data are arranged in order: 5, 6, 6, 6, 8, 8, 9, 9 There are an even number of data points. middle numbers Add the two middle numbers. Divide the sum by 2 to find the median. 6 8 14 14 2 7 The mode is the number that occurs most often: 5, 6, 6, 6, 8, 8, 9, 9. There may be more than one mode or no mode at all. The median is 7, the mode is 6, and the range is 4. Find the median, mode, and range of each data set. 1. Julia s ice skating scores: 9, 8, 8, 7, 8, 6, 7, 8, 6 Median: [8] Mode: [8] Range: [3] 2. Calvin s skateboarding scores: 4, 8, 5, 7, 8, 8, 7, 5 To find the range, subtract the least number from the greatest number. 9 5 4 Median: [7] Mode: [8] Range: [4] 3. Class heights in inches: 53, 48, 46, 51, 49, 52 Median: [50] Mode: [none] Range: [7] Discovering Math, Statistics and Data Analysis, Summarizing Data...or, Around the Middle 2
SUMMARIZING DATA The mean, median, mode, and range are used to summarize data. The mean, the median, and the mode indicate central or typical data. The range shows the spread of the data. At a diving practice, Nadia received the following scores from a number of teammates: 9, 6, 8, 6, 6, 7 Find the mean, median, mode, and range of the scores. The mean is the average of the numbers in the data set. To find the mean, add the numbers in a data set. 9 6 8 6 6 7 42 Then divide the sum by the number of addends. 42 6 7 Put the scores in order from least to greatest: 6, 6, 6, 7, 8, 9. The median is the middle number in the data set. 6, 6, 6, 7, 8, 9 There are an even number of data points. Add the two middle numbers. Divide the sum by 2 to find the median. 6 7 13 13 2 6.5 The mode is the number that occurs most often. 6, 6, 6, 7, 8, 9. To find the range, subtract the least number from the greatest number. There may be more than one mode or no mode at all. 9 6 3 The mean is 7, the median is 6.5, the mode is 6, and the range is 3. Find the mean, median, mode, and range of each data set. 1. Melanie s snowboarding scores: 9, 8, 5, 7, 10, 9 Mean: [8] Median: [8.5] Mode: [9] Range: [5] 2. Calvin s diving scores: 6, 9, 5, 8, 6, 9, 7, 6, 7 Mean: [7] Median: [7] Mode: [6] Range: [4] 3. Math quiz scores: 84, 88, 94, 85, 82 Mean: [86.6] Median: [85] Mode: [none] Range: [12] Discovering Math, Statistics and Data Analysis, Summarizing Data...or, Around the Middle 3
GRAPHING DATA The table shows the favorite pets of a group of third graders. You can make a bar graph to display and interpret the data. Favorite Pets Pet Cat Dog Ferret Fish Bird Number of Students 8 10 3 5 2 To make a bar graph of the data in the table: 12 FAVORITE PETS Step 1 Step 2 Step 3 Step 4 Draw and label the side and bottom of the graph. Write the title at the top. Start with 0 and choose a scale. Here, multiples of 2 allow it to be clear and not too crowded. Write the numbers along one side. List the pets to label the bars. Draw bars to match the data. From the graph, you can see quickly that the most popular pet was dogs. Use the data in the table to make a bar graph. Number of Students 10 8 6 4 2 0 Cat Dog Ferret Fish Bird Pet Favorite Sport Sport Soccer Skateboard Cycling Basketball Softball Number of 8 4 8 6 3 Students N umber of Students 10 8 6 4 2 0 FAVORITE SPORT Soccer Skateboard Cycling Sport Basketball Softball Discovering Math, Statistics and Data Analysis, Graphing Data...or, Charting a Course 1
GRAPHING DATA The table shows the belt levels of students in a karate club. You can make a bar graph to display and interpret the data. Color Number of Students Belt Levels White Yellow Green Blue Brown Black 25 20 25 15 10 5 To make a bar graph of the data in the table: Step 1 Step 2 Step 3 Step 4 Draw and label each axis. Write the title at the top. Start with 0 and choose a scale. Here intervals of 5 allow it to be clear and not too crowded. Write the numbers along one axis. List the belt colors to label the bars, on the other axis. Draw bars to match the data. From the graph, you can see quickly which belts are held by more students and which by less. Use the data in the table to make a bar graph. Number of Students 30 25 20 15 10 5 0 BELT S White Yellow Green Blue Brown Black Belt Color Favorite Activity Activity Shop Movies Sports Video Read Games Number of Students 10 10 25 15 5 N umber of Students FAVORITE WEEKEND ACTIVITY 30 25 20 15 10 5 0 Shop Movies Sports Video Read Games Activity Discovering Math, Statistics and Data Analysis, Graphing Data...or, Charting a Course 2
GRAPHING DATA The table shows the amount of money raised at a School Book Fair from 2001 through 2005. You can make a line graph to show changes over time. To make a line graph of the data in the table: Step 1 Step 2 Step 3 Step 4 Step 5 Book Fair Sales Year 2001 2002 2003 2004 2005 Money Raised $200 $400 $450 $600 $650 Draw and label the axes. Title the graph. Start with 0 and choose a scale for the numerical axis. Here, intervals of 100 allow it to be clear and not too crowded. Write the numbers along the vertical axis. List the years along the horizontal axis. Plot the points on the graph. Connect the points in order. Money Raised (dollars) 700 600 500 400 300 200 100 0 BOOK FAIR SALES 2001 2002 2003 2004 2005 Year From the graph, you can see that the money raised has increased every year. Use the data in the table to make a line graph. Magazine Sales Year 2000 2001 2002 2003 2004 Number 75 125 100 150 175 Sold Number Sold 200 175 150 125 100 75 50 25 0 MAGAZINE SALES 2000 2001 2002 2003 2004 Year Discovering Math, Statistics and Data Analysis, Graphing Data...or, Charting a Course 3
READING GRAPHS Graphs are visual ways to show data. A bar graph uses bars to display the data. The bar graph shows the pets of third graders at Washington Elementary School. You can use the graph to interpret the data. The bar for dogs is the longest bar in the graph. It shows that more students have dogs than have any other type of pets. The bar for birds shows that 8 students have birds for pets. Number of Students 30 20 10 0 OUR PETS Dogs Cats Fish Birds Horses Other Type of Pet Use the bar graph to answer each question. 1. How many students have cats for pets? [18 students] 2. How many more students have dogs than cats for pets? [2 students] 3. Which two types of pets do the same number of students have? [fish and horses] 4. Natalie has a pet ferret. Which bar on the graph includes her pet? [the bar labeled other ] 5. How many students have birds or fish for pets? [13 students] 6. Do more students have cats or horses for pets? How do you know? [Cats; because the bar for cats is taller than the bar for horses.] 7. Do more students have birds or fish for pets? How do you know? [Birds; because the bar for birds is taller than the bar for fish.] CHALLENGE How many third graders at Washington Elementary have pets, if each student with pets has only one kind of pet? [66 students] Discovering Math, Statistics and Data Analysis, Reading Graphs...or, Story Lines 1
READING GRAPHS Graphs are visual ways to show data. A circle graph displays data on how parts of a set, a quantity, or a thing make up the whole set, quantity, or thing. The circle graph shows how a class spends their 1 hour PE class. You can use the graph to interpret the data. The graph shows that about 1 4 of the time is spent building strength. One quarter of an hour is 15 minutes. So the class spends about 15 minutes building strength. OUR 1 HOUR PE CLASS Aerobics Build Strength Warm-up Cool down Use the circle graph to answer each question. 1. Which activity takes up the greatest part of the class? Explain. [Aerobics, because it is the greatest part of the graph.] 2. Which two activities take about the same amount of time? [warm-up and cool down] 3. Is more time spent building strength or cooling down? [building strength] 4. About how much time is spent doing aerobics? Explain. [About 30 minutes since about half the graph is for aerobics and half of 1 hour is 30 minutes.] 5. About how much time is spent doing aerobics and building strength? Explain. [About 45 minutes since 30 min + 15 min = 45 min] CHALLENGE Is it reasonable to conclude that about 15 minutes of class time are spent on the warm-up from the graph? Explain. [No. Possible answer: Together the warm-up and cool down make up about 1 4 of the graph and one quarter of an hour is 15 minutes. So the time must be less than 15 minutes.] Discovering Math, Statistics and Data Analysis, Reading Graphs...or, Story Lines 2
READING GRAPHS Graphs are visual ways to show data. A line graph can be used to show changes over time. The line graph shows the high temperature recorded each day for a 7-day period. You can use the graph to interpret the data. The graph shows that the high temperature on Monday was 78 F. It also shows that the high temperature increased from Monday to Tuesday. Temperature ( F) 90 88 86 84 82 80 78 76 74 72 70 0 HIGH TEMPERATURES Mon. Tues. Wed. Thurs. Fri. Sat. Sun. Day Use the line graph to answer each question. 1. What was the high temperature on Thursday? [86 F] 2. Was the high temperature greater on Tuesday or on Thursday? How do you know? [Thursday; because the point for Thursday is higher on the graph than the point for Tuesday.] 3. How did the high temperature change from Wednesday to Thursday? [it increased by 2 F] 4. How much warmer was the high temperature on Friday than on Monday? [9 warmer on Friday] 5. Between which two days was there no change in high temperature? How does the graph show this? [Between Friday and Saturday. The line between the points for these days is horizontal.] 6. What pattern does the graph show in the high temperatures over the week? [the temperatures increased over the week] Discovering Math, Statistics and Data Analysis, Reading Graphs...or, Story Lines 3
COLLECTING AND HANDLING DATA Different types of graphs can be used to display different types of data. Line Graph A line graph shows changes of one variable with another, often over time. Use a line graph to show the weight gain of a puppy during its first year. Weight (pounds) 25 20 15 10 5 0 WEIGHT OF A PUPPY J F M A M J J A S O N D Month Bar Graph A bar graph displays data for categories or groups, for easy comparison. Use a bar graph to compare classmates favorite colors. Number of Students 12 10 8 6 4 2 0 FAVORITE COLORS Blue Red Green Yellow Color Circle Graph A circle graph displays the parts of a whole, for easy comparison. Use a circle graph to show how you spend each part of a typical day. School MY DAY Meals Sleep Sports Homework Which type of graph would you use to display each data set? Write line graph, bar graph, or circle graph. 1. Show the number of inches a baby grows during its first year. 2. Show what a child does with different portions of all the birthday money. [circle graph] 3. Show how many third-graders have different favorite breakfast foods at a school. [bar graph] 4. Show the amount of rainfall each month for 6 months. 5. Show number of students with different belts at a karate school. 6. Show how time is divided among all the activities of a science class. [circle graph] [line graph] [line graph or bar graph] [bar graph] 7. Show how the height of a plant changes during the year. [line graph] Discovering Math, Statistics and Data Analysis, Collecting and Handling Data...or, Form Counts 1
COLLECTING AND HANDLING DATA Different types of graphs can be used to display different types of data. Line Graph A line graph shows changes of one variable with another, often with time. Use a line graph to show the low temperature each day for 5 days. Bar Graph A bar graph displays data for categories or groups, for easy comparison. Use a bar graph to compare attendance at a book fair for 4 days. Circle Graph A circle graph displays the parts of a whole, for easy comparison. Use a circle graph to show how you spend your allowance each month. MY ALLOWANCE Entertainment Savings Food Games Which type of graph would you use to display each data set? Write line graph, bar graph, or circle graph. 1. Show the weight gain of a baby during its first year. [line graph] 2. Show how a child spends a typical day. [circle graph] 3. Show numbers of different types of rocks in a rock collection. [bar graph or circle graph] 4. Show the numbers of home runs of different baseball players. [bar graph] 5. Show the number of skateboards sold each year for 5 years. [bar graph or line graph] 6. Show how time is spent at a typical camp day. [circle graph] 7. Show how the height of an office building changes as it is being built. [line graph] 8. Show the different types of books in a library. [circle graph] Discovering Math, Statistics and Data Analysis, Collecting and Handling Data...or, Form Counts 2
COLLECTING AND HANDLING DATA Different types of graphs can be used to display different types of data. Double line and double bar graphs show two sets of data on one graph, making comparisons easy. Both types of graphs have keys that distinguish the sets of data. Double Line Graph A double line graph compares two different sets of data on one graph. Each line shows how the value of one variable changes with another, often with time. Double Bar Graph A double bar graph shows two different sets of data on one graph. It compares data by categories. Which type of graph would best display each data set? Write double line graph, double bar graph or either. 1. Show daily number of DVD rentals at two different stores over a 4-week period. [either] 2. Show attendance of boys and girls at a carnival from Thursday through Sunday. [either] 3. Show average lengths of males and females of different species of dolphins. [double bar graph] 4. Show temperature each hour for 2 different cities over an 8-hour period. [double line graph] 5. Show costs of two different brands of skateboards in 2000 and in 2005. [double bar graph] 6. Show monthly savings, (month by month) of a brother and sister for 6 months. [either] Discovering Math, Statistics and Data Analysis, Collecting and Handling Data...or, Form Counts 3