Materials Needed: Centimeter Rulers, Hands, Activity Worksheet (See Below)

Teacher Instructions and Activity Description for measures of spread

Materials Needed: centimeter rulers, hands, activity worksheet (see below)

Measure the handspan of each individual in the classroom in centimeters. Share the data.

Discussion:

· Why would you summarize data into a single number (a statistic)?

· How do you summarize data into a single number?

· What does it mean to be average?

· What are the different measures of center? mean, median

· Is there a way to determine the most appropriate measure of center? Or, why might a particular measure of center be more useful in a given situation? Mean is only useful for data that is symmetric. It changes greatly when there are outliers. Example: If Bill Gates moved to Logan, the mean income would increase so that the average income would be in the millions, however, the only multiple billionaire would be Bill. Median is useful for most data sets.

· What are different measures of variation (spread)? standard deviation, inter-quartile range

· How do the various measures of variation (spread) relate to the measures of center? Mean—standard deviation and median----IQR

· Why is it important to know measures of variation and how are these measures used?

· What is the range of the hand-span data? highest value – lowest value

· Arrange the data into lowest to highest (do in graphing calculator using lists).

· Find the 5 number summary: minimum, quartile 1, median, quartile 3 and maximum. Remember that quartile 2 = median.

· Make a box and whisker plot. Do again using the graphing calculator. Use the trace feature to locate the 5 number summary.

· Find the percentiles: divide the data into 10ths.

· Discuss what handspans are in the 90th percentile, 75th percentile, ect.

Name ______Class Period______

Standard Deviation

19 Steps to Understanding the Standard Deviation

Fill in the blanks and answer the following questions to learn about standard deviations.

1. The mean,, of a set of n data values, x1, x2, x3, … xn , is computed by summing [you can use the Greek letter “s”, ∑ ] all of the data values and then dividing the sum by ______.

Symbolically write the formula for the mean, :

Note: Statisticians often use a shortened form of this formula that does not include the increment values on the “∑”, since statisticians know they will usually be adding up all of the data values from 1 to “n” for their data sets.

Symbolically write the shortened formula for the mean, :

2. Get a ruler and measure your hand span (the distance from the tip of your thumb to the tip of your little finger when you spread your fingers). Measure to the nearest half centimeter.

My hand span is about ______centimeters.

3. Form a group of four statistics students. Record the hand spans of the other three members of your group.

______’s hand span is ______centimeters.

4. On the number line below, make a dot plot of the hand spans of the four members of your group. Write your initials above the dots to identify the members of your group.

5. Find the mean hand span for your group. Our mean hand span is ______cm.

6. Mark the mean hand span on your dot plot with a wedge [▲] below the number line.

7. How far from the mean are the hand spans of your group? When computing the difference from the mean, all hand spans smaller than the mean will have negative values and all hand spans larger than the mean will have positive values.

How far from the mean are the hand spans of the members of your group?

My hand span is ______centimeters from the mean.

______'s hand span is ______centimeters from the mean.

8. On the number line below, make a dot plot of your group’s distances from the mean.

9. These differences are represented symbolically as , and are called deviations from the mean, or just deviations.

10. What is the sum [ ∑ ] of the deviations from the mean for your group? ______cm.

Symbolically write the formula for the sum of the deviations from the mean:

11. Note that the sum of the deviations (from the mean) always equals ______. Why?

12. To find the “average” deviation from the mean, the values need to be all positive. For various reasons, statisticians DO NOT usually use the absolute value to make the values positive, instead statisticians square the difference from the mean to describe a “standard deviation.” To find a squared deviation, just square your difference (deviation) from the mean, ,

List the squared deviations for the mean for your group.

My squared deviation is ______square centimeters.

______'s squared deviation is ______square centimeters.

13. The sum of the squared deviations from the mean is often called the total sum of the squares, (SST).

Symbolically write the formula for the total sum of the squared deviations from the mean:

SST =

14. What is your group’s total sum of squared deviations from the mean? ______sq. cm.

15. What should you do to find an “average” squared deviation?

The mean of the sum of the squared deviations from the mean is called the variance.

Symbolically write the formula for the variance:

Variance =

16. What is your group’s variance (mean squared deviation)? ______sq. cm.

17. The variance for your hand spans is in square centimeters. For standardizing purposes, you can take the square root of the variance to get values that are in centimeters. The square root of the mean squared deviation, or the square root of the variance, is called the standard deviation.

Symbolically write the formula for the mean squared deviation, or standard deviation, :

[ is the non-capital Greek letter “s”]

Standard deviation, =

The formula above is the formula for a population standard deviation. If a statistician wants to find the standard deviation for a sample, he/she makes an adjustment to the formula. Instead of dividing by n, just use the same formula but divide by n − 1. This adjustment gives more reasonable values for the standard deviation when you are working with a sample, especially a small sample.

Symbolically write the formula for the standard deviation of a sample, s:

Note: As the size of your sample increases, it makes little difference whether you divide by n or n − 1.

Sample standard deviation, s =

18. What is your group’s sample standard deviation? s = ______cm.

19. It is difficult and tedious to calculate standard deviations by hand when you have a lot of data. Find the standard deviation for the entire class using a graphing calculator 1-Variable Statistics command.

Statistics is FUNdamental – 2008