Coins, Presidents, and Justices: Normal Distributions and z-scores

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1 activity 17.1 Coins, Presidents, and Justices: Normal Distributions and z-scores In the first part of this activity, you will generate some data that should have an approximately normal (or bell-shaped) distribution. In the second part, you will use the definition of standard deviation and compare the standard deviations for two different data sets. 1. Work with a partner to generate the following dat Toss 10 coins and record the number of heads you obtained.

2 570 Excel Activities Repeat this 24 more times until you have a list of 25 numbers, each of them between 0 and 10. Retrieve the file EA17.1 Coins and Presidents.xls from the CD or website, and you will find the results of 35 tosses of 10 coins that someone else carried out. When you first retrieve the file, column B contains the number of times 0 heads was obtained in the 35 tosses of 10 coins, the number of times 1 head was obtained in the 35 tosses, and so on, up to the number of times 10 heads was obtained. Add your results to the list so you have a total of 60 in column B. Create a scatterplot of these data, using one of the versions of the scatterplot with the dots connected. Be sure to add axis titles to your scatterplot. Describe what your curve looks like, including where it is centered and what its spread is. e. Change your graph to a bar graph (instructions follow).

3 Activity 17.1: Coins, Presidents, and Justices 571 Instructions to Use Excel to Change a Scatterplot to a Column Graph Click on the plot area and, from the Design tab, select Change Chart Type from the Type group. From the drop-down menu, select Column to change your graph to a column (that is, a bar) graph and click OK. Click one of the bars, go to the Format tab and choose Format Selection from the Current Selection group. From Series Options, change the Gap width to 0, by moving the slide to the left, so adjacent bars will touch. (This will make it look like a histogram, and then you can sketch a curve along the tops of the bars.) Click Close. f. Print your bar graph, with appropriate titles on the axes, and by hand draw in a bell-shaped curve that fits this dat How does your handdrawn curve compare with the curve you described in part d of this question? 2. In the second part of this activity, you will examine one measure of the spread of a data set, the standard deviation. The standard deviation plays an important role in understanding the spread of a distribution, especially a bellshaped or normal distribution. You ll use the data set of ages of U.S. presidents at their inauguration, which can be found on sheet 2 of the file EA17.1 Coins and Presidents.xls, and calculate the standard deviation of this data set. To do this, first find the mean (average) of the ages and store that value in cell B46. (Source: The World Almanac and Book of Facts 2004, page 563.) Record the mean age here:

4 572 Excel Activities In cell C1, enter the label Deviation from the mean. In cell C2, enter an appropriate formula to subtract the mean age from the value in cell B2 and that will allow you to drag down to compute each data value minus the mean. (Remember to use $B$46 to keep the value of the mean fixed when you drag.) Drag down to compute each data value minus the mean. How many of the values in this column are negative? What property makes these values negative? Add the values in column C and record their sum here: Write this number without using scientific notation: This number should be 0 or very, very close to 0. Is it? d. Now you want to compute the square of the deviation from the mean of each data value. In cell D1, enter the label Squared deviation and use the instructions that follow to enter the squares of the deviation values in column D. Then add all the values in column D, store that number in cell D46, and record the sum here:. Instructions to Use Excel to Compute Squares In cell D2, enter the formula =C2^2. Then drag this formula down to cell D44.

5 Activity 17.1: Coins, Presidents, and Justices 573 e. f. Next you need to divide the sum of the squared deviations (the value in cell D46) by one less than the number of data points. There are 43 data points, so divide the value in cell D46 by 42; store this number in cell E46. Finally, compute the standard deviation by taking the square root of the number in cell E46 (see the following instructions for a reminder of how to compute the non-negative square root of a number). Store this value in cell F46 and record it here: standard deviation = Instructions to Use Excel to Compute a Square Root To compute the non-negative square root of the value in E46, use the command =SQRT(E46). g. Check the computations by using the following Excel command to compute the standard deviation of the ages in column B. Enter the value obtained in cell F47, and write this value here:. Instructions to Use Excel to Compute the Standard Deviation To compute the standard deviation of the values in cells B2 through B44, use the command =STDEV(B2:B44).

6 574 Excel Activities h. The standard deviation of a set of data is a measure of the spread of the data values. It incorporates the sum of the squared deviations from the mean, of all data values. Suppose Clinton had been 86 years old instead of 46 at his inauguration. What would change in the computations you just performed? i. Change Clinton s age on your spreadsheet from 46 to 86. Notice that all of your computations change automatically. Record the new mean and the new standard deviation. mean = ; standard deviation = j. Change Clinton s age back to its correct value of Go to sheet 3 of the file EA17.1 Coins and Presidents.xls, where you will find another data set. This data set contains the names and ages at time of appointment as chief justice for all the chief justices of the U.S. Supreme Court. (Source: The World Almanac and Book of Facts 2006, page 53.) Compute the mean and standard deviation of these ages, and record these values here: mean = ; standard deviation = Describe how the means and standard deviations of the two data sets, Presidents Ages and Supreme Court Chief Justices Ages, compare.

7 Activity 17.1: Coins, Presidents, and Justices 575 Pick the maximum data value in the Presidents Ages data set. Call it x mean x, and compute its z-score by computing: z =, using the standard deviation mean and the standard deviation of the presidents ages. Record this z-score here: d. Find the z-score for the largest data value in the Supreme Court Chief Justices Ages data set, using the mean and standard deviation of the chief justices ages. Record this z-score here: e. What do the two z-scores tell you? Summary In the first part of this activity, you generated data and created a graph to see that the data has an approximately normal distribution. In the second part of the activity, you worked with the standard deviation to see the impact of a large data value on this measure of spread. You also compared data from two data sets by looking at z-scores.

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