To Start the Process, This Packet Will Serve As Your Text

PATTERNS IN DATA

LESSON 3 VARIABILITY

TO START THE PROCESS, THIS PACKET WILL SERVE AS YOUR TEXT

YOU MAY WRITE IN THIS PACKET (BACK OF EACH PAGE IS BLANK); YOUR ANSWERS MUST BE COMPLETE, NEAT, AND ORGANIZED! IF YOU DO NOT HAVE ENOUGH ROOM, FEEL FREE TO USE PAPER

1 LESSON 3 VARIABILITY

Whenever two people observe an event, they are likely to see different things. If two people measure something to the nearest millimeter, they probably will get two different measurements. If two people conduct the same experiment, they will get slightly different results. In fact, there is variability in nearly everything; no two leaves or snowflakes are exactly alike. Because variability is so common, it is important that you begin to understand what causes variability and how it can be measured and interpreted.

Think About This Situation

a. Using the plots above, is it reasonable to call a 14-year-old boy “taller than average” if his height is 165 cm? Is it reasonable to call a 14-year-old boy “tall” if his height is 165 cm? What additional information about 14-year-old boys would you need to know to be able to say that he is “tall”?

2 b. At what height would you be willing to call a 14-year-old girl “tall”? Do you have enough information to make this judgment?

c. The average heights come from a physician’s handbook. If you were a physician, what other information would you need to know in order to decide if a child is growing normally? INVESTIGATION 1 Measuring Variability: The Five Number Summary

If you are in the 40th percentile of height for your age, that means that 40% of people your age are shorter than you are and 60% are taller. You have been given copies of physical growth charts for boys and girls, 2 to 18 years in age. The charts were developed by the National Center for Health Statistics.

The curved lines for the height (top) and weight (bottom) tell a physician what percentile a boy or girl is in. The percentiles are the small numbers 5, 10, 25, 50, 75, 90, and 95 towards the right ends of the curved lines. For example, suppose John is a 17-year-old boy who weighs 60 kg (132 pounds). John is in the 25th weight percentile for his age. Twenty-five percent of 17-year- old boys weigh less than John and 75% weigh the same or more than John. John’s height is 180 cm (almost 5’11”). He is in the 75th height percentile for his age.

1. Based on the information given about John, how would you describe John’s general physical appearance?

2. Use the NCHS growth charts to answer the following questions in your group.

a. What is the approximate percentile for a 9-year-old girl who is 128 cm tall?

3 b. About how tall does a 12-year-old girl have to be so that she is taller than 75% of the girls her age?

c. How tall would a 14-year-old boy have to be so that you would consider him “tall” for his age? How did you make this decision?

d. How tall would a 14-year-old girl have to be so that you would consider her “tall” for her age? How did you make this decision? e. What is the 25th percentile in height for 4-year-old boys? The 50th percentile? The 75th percentile?

f. How can you tell from the height and weight chart when (which ages) children are growing the fastest? When is the increase in weight the greatest for girls? For boys?

a. What is another name for the 50th percentile? (Think measures of center)

b. The 25th percentile is sometimes called the lower quartile. Estimate the lower quartile of height for 6-year-old girls.

4 c. The 75th percentile is sometimes called the upper quartile. Estimate the upper quartile of height for 6-year-old girls.

The quartiles together with the median give some indication of the center and spread (variability) of a set of data. A more complete picture of the distribution of a set of data is given by the five- number summary, which includes: the minimum value, the lower quartile (Q1), the median (Q2), the upper quartile (Q3), and the maximum value.

4. From the charts, estimate the five-number summary for 13-year-old girls’ heights and for 13-year-old boys’ heights. Some estimates will be more difficult than others.

The distance between the first and third quartiles is called the interquartile range (IQR). The IQR is a measure of how spread out the data is. The distance between the minimum value and the maximum value is called the range. The range is another, typically less useful, measure of how spread out the data is.

a. What is the interquartile range of the heights of 13-year-old girls? Of 13-year-old boys?

b. Based on the NCHS charts, what happens to the IQR of heights as children get older?

5 c. In general, do boys’ heights or girls’ heights have the larger interquartile range or are they about the same?

d. What happens to the IQR of weights as children get older?

a. Based on the NCHS charts, is it possible to estimate the range of heights of 18-year-old boys? Explain why or why not.

b. Why is the interquartile range more informative than the range?

For the children’s heights, you were able to estimate quartiles from the chart. Next you will learn how to compute them from sets of data.

When you explored the AAA ratings for cars in Lesson 2, Investigation 1, you saw variability in the ratings of each car. Below are the ratings for the Civic del Sol and the Laser and their corresponding histograms.

6 7. Which of the cars has greater variability in its ratings? Explain your reasoning.

a. Put the ratings for the Civic del Sol in an ordered list from smallest to largest. Find the median. Mark the position of the median in your list.

b. How many ratings are on each side of the median?

c. Once the data is ordered, you can divide them into quarters to find the quartiles. Find the midpoint of each half of the ratings for the Civic del Sol. Mark the position of the quartiles (lower and upper) in your list.

d. What fraction or percentage of the ratings is less than or equal to the lower quartile?

7 e. What fraction or percentage of the ratings is greater than or equal to the upper quartile?

a. You can find the five-number summary quickly using technology. Find the five- number summary for the Laser’s ratings by calculating the one-variable statistics on your calculator.

b. What is the 25th percentile of the ratings for the Laser? The 75th percentile?

8 INVESTIGATION 2 Picturing Variability

You will need your class data table from Lesson 1, Investigation 1 for these problems.

1. Find the five-number summary of the class data for arm span using your calculator.

The five-number summary can be displayed in a box plot. To make a box plot of your arm span data, first make a number line. Below this line draw a box from the lower quartile to the upper quartile; then draw line segments connecting the box to each extreme (the maximum and minimum values). Draw a vertical line in the box to indicate the median. The segments at either end are often called whiskers, and the plot is sometimes called a box-and-whiskers plot. Here are the results for one class (in inches).

a. Is the distribution represented by the sample box plot above skewed to the left or to the right, or is it symmetric?

b. Draw a box plot of the arm span data for your class.

9 c. Is the distribution for your class skewed to the left, right or is it symmetric?

d. What is the length of the box for your class data? What is the mathematical term for this length?

Box plots are most useful when comparing two or more distributions. You can produce box plots with your graphing calculator by following a procedure similar to that of making histograms. After entering the data and specifying the viewing window, select Box Plot as the type of graph desired.

3. Refer to the ratings for the Civic del Sol and the Laser given in Investigation 1 on page 6.

a. On your calculator, make a box plot of the ratings. You should have the box plot for both cars graphed at the same time. You do not need to copy a picture of the box plots in your notes.

b. What do the different lengths of the boxes tell you about the variability in their AAA ratings? Is either distribution skewed? Is either distribution symmetric?

c. Use the trace feature to find the five-number summary for both cars. Compare the results with your computations in #8 from Investigation 1 of this lesson.

10 d. Compare the box plots for the Civic del Sol and the Laser. You should include important characteristics such as a comparison of a measure of center you can see on the box plots and the two measures of spread (variability) you can see from the box plots.

Reproduced below are the ratings for the LeBaron, Camaro, and Miata.

4. Divide the following problem amongst your group members. Produce box plots for the LeBaron, Camaro, and Miata ratings. Each person needs to only copy one box plot into the notes.

a. Use the box plots to determine which of the three cars has the largest IQR.

b. Including the Civic del Sol and the Laser, which cars have almost the same IQR? Does this mean their distributions are the same? Explain your reasoning.

11 c. Why does the Miata have no whisker at the upper end?

d. Why is the lower whisker for the Miata so long?

e. Based on the box plots, which of the three cars seems to have the best ratings? How did the plots help you make your decision?