# Measures of Data Spread

Refer to the below data sets of scores, X, Y and Z, from two 5-point quizzes to answer the questions below.

X f Y f Z f

5 4 5 1 5 0

4 3 4 2 4 0

3 0 3 4 3 1

2 1 2 1 2 4

1 1 1 1 1 1

0 1 0 1 0 4

#1. Compute the range of scores for data sets (a) X, (b) Y and (c) Z.

#2. For X data set, compute the interquartile range and the SIQR, also known as the semi-interquartile range.

? Why is it desirable to use SIQR as the measure of variability?

? Note: In order to compute the interquartile range, we must first find both the score at the 25th and the 75th percentile.

#3. Using the X data set of scores above, compute the following:

? The variance and standard deviation, supposing the data sets above are from a known population (the entire population is tested).

? Then, the variance and standard deviation, supposing that the data sets above are used to estimate the population values.

#4. Using the Y data set of scores above, compute the following:

? The variance and standard deviation, supposing the data sets above are from a known population (the entire population is tested).

? Then, the variance and standard deviation, supposing that the data sets above are used to estimate the population values.

#5. Using the Z data set of scores above, compute the following:

? The variance and standard deviation, supposing the data sets above are from a known population (the entire population is tested).

? Then, the variance and standard deviation, supposing that the data sets above are used to estimate the population values.

#6. (a) When computing the variance and standard deviation of a set of scores, when do we divide by N, and when do we divide by N-1? (b) What is the difference between  and s?

#7. When determining the variability of a set of scores, why is it necessary to square the deviations from the mean, rather than simply use the sum of the deviations from the mean?

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#### Solution Preview

Instructions: Complete the work below, making any necessary computations on separate sheets of paper, and then use the Chapter 5 Exercise Form to record your answers.

Refer to the below data sets of scores, X, Y and Z, from two 5-point quizzes to answer the questions below.

X f Y f Z f

5 4 5 1 5 0

4 3 4 2 4 0

3 0 3 4 3 1

2 1 2 1 2 4

1 1 1 1 1 1

0 1 0 1 0 4

First of all, here is the data as a list, instead of using frequencies:

X Y Z

5 5 3

5 4 2

5 4 2

5 3 2

4 3 2

4 3 1

4 3 0

2 2 0

1 1 0

0 0 0

#1. Compute the range of scores for data sets (a) X, (b) Y and (c) Z.

The range is the highest score minus the lowest score. For all three sets of data, the range will be 5 - 0 = 5.

#2. For X data set, compute the interquartile range and the SIQR, also known as the semi-interquartile range.

? Why is it desirable to use SIQR as the measure of variability?

? Note: In order to compute the interquartile range, we must first find both the score at the 25th percentile and the 75th percentile?which entails using a procedure learned in Chapter 3 -- called "Case 2"!

You use the interquartile range and/or the SIQR if you want to have a measure of variability that is not affected by extreme values. Just as the median is not affected by extreme values when compared to the mean, the interquartile range is not affected by extreme values when compared to other measures of variation such as the range, ...