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statistical displays

statistical displays. chapter nine. Measures of central tendency 9-1B. The sizes of students’ bicycles are listed in the table. Which number appears most often in the table? Order the numbers from least to greatest. Which number(s) is in the middle?. MEASURES OF CENTRAL TENDENCY

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statistical displays

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  1. statistical displays chapter nine

  2. Measures of central tendency9-1B The sizes of students’ bicycles are listed in the table. Which number appears most often in the table? Order the numbers from least to greatest. Which number(s) is in the middle? MEASURES OF CENTRAL TENDENCY are numbers that describe the center of a data set. They are the mean, median, and mode.

  3. measures of central tendency The number of DVDs rented at Red Box during a week are listed in the table. Find the mean, median, and mode The mean is 57 The median is 55 The mode is 34 The points scored in each game by Darby Middle School’s football team are 21, 35, 14, 17, 28, 14, 7, 21, and 14. Find the mean median, and mode. 19; 17; 14

  4. the effect of extreme values The number of songs downloaded during one week are shown in the table. Which measure is MOST affected by Saturday’s downloads? So… find the mean, median, and mode WITH Saturday’s downloads Then…find the mean, median, and mode WITHOUT Saturday’s downloads With Saturday: Mean: 8 Median: 4 Mode: None Without Saturday: Mean: 3.33 Median: 3.5 Mode: None So, how do Saturday’s downloads affect the measures of central tendency? Which one is affected the most?

  5. the effect of extreme values The number of library books returned at Edison Middle School is shown in the table. Which measure is most affected by Thursday’s returned books? Mean The number of students in class during one week is shown in the table. Which measure is most affected by Tuesday’s attendance? Mean: 19.6 and 21.5 Median: 21 and 21.5 Mode: no mode

  6. When are they useful? The following set of data shows the number of push-ups Kris did in one minute for the past 5 days: 24, 21, 28, 27, and 26. Which measure of central tendency best represents the data. JUSTIFY your answer and find the measure! MEAN! 25.2

  7. Measures of central tendency The following set of data shows the costs of pairs of jeans for various stores: $25.99, $29.99, $34.99, $19.99, $45.99. Which measure of central tendency best represents the data. Justify your response and then find that measure. Median because $45.99 is an extreme; $29.99 The following set of data shows the number of miles Jodie ran for the past six days: 4, 5, 4, 4, 6, 4. Which measure best represents the data? Justify your response and then find the measure. Mode because four of the six numbers are identical. 4 Self Assessment: Complete p. 493 #2 & 4 on your own. Then, check with a partner.

  8. Measures of variation9-2A Jamie asked her classmates how many glasses of water they drink on a typical day. What is the median of the data set? Organize the data into two groups: the top half and the bottom half. How many data values are in each group? What is the median of each group? Find the difference between the two numbers. 2 8 1; 3.5 2.5

  9. Measures of variation MEASURES OF VARIATION are used to describe the distribution of the data QUARTILES are the values that divide the data set into four equal parts LOWER QUARTILE median of the lower half of the data set UPPER QUARTILE median of the upper half of the data set

  10. measures of variation UPPER AND LOWER QUARTILES The upper and lower quartiles are the MEDIANS of the upper half and lower half of a set of data, respectively. INTERQUARTILE RANGE The range of the middle half of the data. It is the difference between the upper quartile and the lower quartile. In this case, 29-18 or 11. RANGE is the difference between the greatest and least data values; in this case 31-14 or 17

  11. find measures of variation Find the measures of variation for the data set. Range Quartiles (L.Q., median, U.Q.) Interquartile Range Range: 69 Median: 27.5 Lower Quartile: 8 Upper Quartile: 50 Interquartile Range: 42

  12. Find measures of variation Range: 15 Median: 60.5 Lower Quartile: 64 Upper Quartile: 58 Interquartile Range: 6 Determine the measures of variation for the data in the tables. Range: 85 Median: 59 Lower Quartile: 41 Upper Quartile: 100 Interquartile Range: 59

  13. find outliers! OUTLIER a data value that is either much greater or much less than the median. IF a data value is more than 1.5 times the value of the interquartile range beyond the quartiles, it is an outlier. The ages of candidates in an election are: 23, 48, 49, 55, 57, 63, and 72. Name any outliers in the data. It’s helpful to get this first: Range: 49 Median: 55 Lower Quartile: 48 Upper Quartile: 63 Interquartile Range: 15 • Steps: • Find the interquartile range • 63-48 = 15 • Multiply the interquartile range by 1.5 • 15 x 1.5 = 22.5 • Subtract 22.5 from the lower quartile and also add 22.5 to the upper quartile. • 48-22.5 = 25.5 63 + 22.5 = 88.5 So…if there were an outlier, it would be lower than 25.5 and higher than 88.5. Therefore 23 is the only outlier.

  14. Find outliers The lengths, in feet, of various bridges are: 88, 251, 275, 354, and 1,121. Name any outliers in the data set. Steps: Find the interquartile range Multiply the interquartile range by 1.5 Subtract 22.5 from the lower quartile and also add 22.5 to the upper quartile. It’s helpful to get this first: Range: Median: Lower Quartile: Upper Quartile: Interquartile Range: NO OUTLIER!!! Try this one: The average daily temperatures in degrees Fahrenheit for one week in July were 94, 92, 90, 95, 71, 89, and 92. Name the outliers in the data. 71

  15. analyze data The table shows a set of scores on a science test in two different classrooms. Compare and contrast their measures of variation. (Find the measures of variation for both rooms.) When you “analyze”, you need to write MULTIPLE sentences describing the likenesses and differences between the measures!

  16. analyze data Temperatures for the first half of the year are given for Antelope, Montana and Augusta, Maine. Compare and contrast the measures of variation of the two cities. It’s helpful to get this first: Range: Median: Lower Quartile: Upper Quartile: Interquartile Range: Range: 58 and 47 Median: 50 and 47 Lower Quartile: 30 and 32 Upper Quartile: 70 and 66 Interquartile Range: 40 and 34 You could say: The medians are close, but the data are more spread out in the Antelope data Self Assessment: Complete p. 500 #1-2 on your own. Then, check with a partner.

  17. box-and-whisker plots9-2b The line plot shows the number of touchdowns scored by each of the 16 teams in the National Football Conference in a recent year. Find the median, quartiles, and range of the data. What percent of the teams scored less than 30 touchdowns? What percent of the teams scored more than 37 touchdowns? Median: 37.5; UQ: 45.5; LQ: 31; Range: 30 25% 50% The medians and quartiles divide the data into four equal parts.

  18. box-and-whisker plot A BOX-AND-WHISKER PLOT is a diagram that is constructed using the median, quartiles, and extreme values. A BOX is drawn around the quartile values. The WHISKERS extend from each quartile to the extreme values. The MEDIAN is marked with a vertical line. You may think that the median always divides the box in half. However, the median may not divide the box in half because the data may be clustered towards one quartile. (Whiskers, the cat!)

  19. box-and-whisker plot You can see all the pieces of data in a LINE PLOT. In a BOX-AND-WHISKER PLOT, you only see the MEDIAN, QUARTILES, and EXTREME VALUES. BOX-AND-WHISKER PLOTS separate data into four parts. Even though the parts may differ in length, EACH CONTAINS 25% OF THE DATA!!!!

  20. box-and-whisker plot The list below shows the speeds of eleven cars. Draw a box-and-whisker plot of the data! 25 35 27 22 34 40 20 19 23 25 30 Step 1: Order the numbers from least to greatest and draw a number line that is appropriate for the range. Step 2: Find the median, the extremes, and the upper and lower quartiles. Mark these points above the number line. Step 3: Draw the box so that it includes the quartile values. Draw a vertical line through the box at the median value. Extend the whiskers from each quartile to the extreme data points.

  21. box-and-whisker plot Draw a box-and-whisker plot of the data below: {$20, $25, $22, $30, $15, $18, $20, $17, $30, $27, $15} Step 1: Order the numbers from least to greatest and draw a number line that is appropriate for the range. Step 2: Find the median, the extremes, and the upper and lower quartiles. Mark these points above the number line. Step 3: Draw the box so that it includes the quartile values. Draw a vertical line through the box at the median value. Extend the whiskers from each quartile to the extreme data points. The list below shows the speed, in miles per hour, of commercial airliners. Draw a box-and-whisker plot of the data. 540, 460, 520, 350, 500, 480, 475, 525, 450, 515 If the data includes outliers, then the whiskers will NOT extend to the outliers, just to the previous data point. Outliers are represented with an asterisk.

  22. interpret the data It is important that you can draw some conclusions when you look at box-and-whisker plots! Think back about the plot with the drivers… Half of the drivers were driving faster than what speed? 25 mph What does the box-and-whisker plot’s length tell about the data? The length of the left half of the box-and-whisker plot is short. This means that the speeds of the slowest half of the cars are concentrated. The speeds of the fastest half of the cars are spread out. What percent were driving faster than 34 miles per hour? 25%

  23. interpret the data Speed of Commercial Airliners Half of the commercial airliners travel faster than what speed? 490 mph The length of the left whisker is short but has an outlier at 350 mph. This means that the speeds of the slowest 25% of the airliners are spread out. Both boxes are the same size, so the middle 50% of the speeds are more concentrated. What does the box-and-whisker plot’s length tell about the data?

  24. Interpret double box-and-whisker! The double box-and-whisker plot below shows the daily attendance of two fitness clubs. Compare and contrast the range and variance of the attendance at Super Fit versus Athletic Club. Super Fit had an attendance between 53 and 72.5. The Athletic Club had an attendance between 57 and 110. The attendance at the Athletic Club varies more than the attendance at Super Fit. The number of games won in each conference of the National Football League is displayed. Compare and contrast the range and variance of each conference. Self Assessment: Complete p. 506 #1-2 on your own. Then, check with a partner.

  25. circle graphs9-3b The students at Pine Ridge Middle School were asked to identify their favorite vegetable. This table shows the results of the survey. Explain how you know that each student selected only one favorite vegetable. If 400 students participated in this survey, how many students preferred carrots? The percents add up to 100%; 180 We would use a CIRCLE GRAPH to display this data because it shows data as parts of a whole and in it, percents add up to 100!

  26. circle graphs We are going to display the data from the example in a circle graph! There are 360 in a circle. Determine what part of the circle will represent each percent from the table. • 45% of 360° = 0.45 x 360° = 162° • 23% of 360° = 0.23 x 360° = 83° • 17% of 360° = 0.17 x 360° = 61° • 15% of 360° = 0.15 x 360° = 54° Draw a circle with a radius as shown. Then use a protractor to draw the first angle, in this case 162. Repeat this step.

  27. Construct a circle graph The table shows the present composition of Earth’s atmosphere. Display the data in a circle graph. • 78% of 360° = 0.78 x 360° = 281° • 21% of 360° = 0.21 x 360° = 76° • 1% of 360°= 0.01 x 360° = 4° • TOTAL: 361° (so cut on one of them)

  28. construct a circle graph When constructing a circle graph, you first may need to convert the data to ratios and decimals and then to degrees and percents. The table shows endangered species in the United States. Make a circle graph of the data. UH OH!!!! 68 + 77 + 14 + 11 = 170 IT’S OVER 100! Find the ratio that compares each number with the total. Write the ratio as a decimal rounded to the nearest hundredth.

  29. Animal species graph continued • 0.4 x 360° = 144° • 0.45 x 360° = 162° • 0.08 x 360° = 29° • 0.06 x 360° = 22° • TOTAL: 357° (so add a smidge on each)

  30. Construct a circle graph The number of Winter Olympic medals won by the U.S. team from 1924 to 2006 is shown in the table. Display the data in a circle graph. • 0.36 x 360° = 130° • 0.37 x 360° = 133° • 0.08 x 360° = 97° • TOTAL: 360° exactly

  31. Analyze a Circle Graph The graph shows the percent of automobiles registered in the western United States in a recent year. 1.) Which state had the most registered automobiles? 2.) If 24 million automobiles were registered in these states, how many more automobiles were registered in California than in Oregon? 3.) Which state had the least number of registered automobiles? California 17.28 million Nevada 4.65 million 4.) What was the total number of registered automobiles in Washington and Oregon?

  32. histograms9-3C Kylie researched the average ticket prices for NBA basketball games for 30 teams. The frequency table shows the results. What do you notice about the price intervals in the table? How many tickets were at least $20 but less than $50? We can display data from a frequency table as a HISTOGRAM! A HISTOGRAM is similar to a bar graph and is used to display numeric data that have been organized into equal intervals.

  33. construct a histogram Choose intervals and make a frequency table of the data shown. Then construct a histogram to represent the data. FREQUENCY TABLE TO DO: Step 1: Figure out the greatest and least values. Step 2: Figure out an appropriate interval. Step 3: Create the 3-column Frequency Table and use tally marks to fill it out!

  34. construct a histogram HISTOGRAM TO DO: Step 1: Draw and label a horizontal and vertical axis. Step 2: Make a TITLE Step 3: Show the intervals from the table on the axis. Step 4: Draw the bars…they should be of equal width and should touch.

  35. construct a histogram The list shown gives a set of test scores. Choose intervals and make a frequency table and construct a histogram to represent the data. • Remember: • Find the least and greatest; choose the interval • Make the frequency table • Construct the histogram with bars of equal width that touch

  36. construct a histogram The list shows the number of milligrams of caffeine in certain types of tea. Use the intervals 1-20; 21-40; 41-60; 61-80; 81-100 to make a frequency table. Then construct a histogram. 8 47 19 34 30 10 58 20 39 32 12 4 22 40 92 18 85 26 27 • Remember: • Find the least and greatest; choose the interval • Make the frequency table • Construct the histogram with bars of equal width that touch

  37. analyze a histogram How many Arizona Diamondbacks players were at bat at least 400 times in a season? 1 + 4 + 2 = 6; so 6 players What percent of the players were at bat 199 times or fewer? 39 players were at bat total. 28 players were at bat 199 times or fewer. 28/38 = 0.72 or 72% What was the greatest number of times at bat for any one player? This cannot be determined! The histogram only tells us that the greatest number of times was somewhere between 600-699 Based on the data above, how many times is an Arizona Diamondbacks player most likely to be at bat? 0-99

  38. analyze a histogram Use the histogram to answer the following questions: How many months had six or more days of rain? What percent of the months had 3 days of rain or less? four about 58% Self Assessment: Complete p. 522 #1-2 on your own. Then, check with a partner.

  39. stem-and-leaf plots9-3E The table shows the average mass in grams of sixteen different species of chicks. Which mass is the lightest? How many of the masses are less than 10 grams? STEM-AND-LEAF PLOT: Data are organized from least to greatest. LEAVES are the the digits of the least place value STEMS are the next place-value digits

  40. how to make a stem-and-leaf plot Step 1: Choose stems using digits in the tens place, 0, 1, and 2. The least value, 5, has 0 in the tens place. The greatest value, 25, has 2 in the tens place. Step 3: Order the leaves and write a key that explains how to read the stems and leaves. INCLUDE A TITLE! Step 2: List the stems from least to greatest in the stem column. Write the leaves, the ones digits, to the right of the corresponding stems. The ones digits of the data form the leaves. Always write each leaf, even if it repeats. The tens digits of the data form the stems. Include a key!

  41. Make your own stem-and-leaf The number of minutes the students in Mr. Blackwell’s class spent doing their homework one night is shown. Display the data in a stem-and-leaf plot. • Think About These: • What do you do when there is no value for the stem? • What happens when a zero is in the ones place? • Why is a S-A-L the best display for this data? Display the data in a stem-and-leaf plot.

  42. analyze a stem-and-leaf plot The stem-and-leaf plot shows the number of chess matches won by members of the Avery Middle School Chess Team. Find the range, median, and mode of the data. Range: 53 Median: 35 Mode: 40 Find the range, median, and mode of the data from Example 1: the chicks. Range: 20 Median: 12 Mode: 12

  43. effect of outliers The measures of central tendency (numbers that describe the center of a data set: mean, median, and mode) can be affected by an outlier! The stem-and-leaf plot shows the number of points scored by a college basketball player. Which measure of central tendency is most affected by the outlier? Well, first…what is the outlier? Mode: The mode, 26, is not affected at all by 43. Calculate the mean and median each without the outlier, 43. Then calculate them including the outlier and compare.

  44. effect of outliers The stem-and-leaf plot shows the number of points scored by a high school basketball player. Which measure of central tendency is most affected by the outlier? The mean Self Assessment: Complete p. 528 #1-3 on your own. Then, check with a partner.

  45. scatterplots and lines of best fit9-4B A SCATTER PLOT shows the relationship between a set of data with TWO VARIABLES graphed as ordered pairs on a coordinate plane. Scatter plots are useful for making predictions because they show trends in data.

  46. identify a relationship Explain whether the scatter plots of data show a positive, negative, or no relationship. Positive Relationship No Relationship Note: In a positive relationship, as the value of x increases, so does the value of y. In a negative relationship, as the value of x increases, the value of y decreases.

  47. use a line to predict The LINE OF BEST FIT is a line that is very close to most of the data points in a scatter plot. It may not touch each one of the points exactly! The graph shows the enrollment at a summer camp for the past several years. Construct a line of best fit. If the trend continues, what will be the enrollment in 2014? To answer this, draw a line that is close to most of the data points. If the trend continues, the enrollment in 2014 will be about 190 campers.

  48. use a line to predict The graph shows the number of hits on a Web site for the first 5 days. Construct a line of best fit. If the trend continues, predict the day the Web site will have 12,000 hits. Day 7 The line graph shows the annual attendance at a county fair over the last 10 years. Construct a line of best fit. If the trend continues, how many people will attend the fair in 2014? about 66,000 people

  49. use a scatter plot to predict The scatter plot shows the earnings for the winning driver of the Daytona 500 from 1996 to 2008. Predict the winning earnings for the 2012 Daytona 500. The predicted winning earnings for 2012 will be about $2,000,000 Predict the winning earnings for 2014! The predicted winning earnings for 2014 will be about $2,250,000

  50. make your own scatter plot Use the table that shows the relationship between hours of sleep and scores on a test Display the data in a scatter plot Describe the relationship, if any, between the two variables Predict the test score for someone that sleeps for 5 hours. 2. Positive Relationship; As the hours of sleep increase, the scores increase 3. About a 65 Self Assessment: Complete p. 536 #1-2 on your own. Then, check with a partner.

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