Chapter 5: Variance

To calculate the standard deviation for a population, you need to use the ________ function (P for population).

STDEVP =STDEVP(A1:A12)

Population Standard Deviation Formula

sum of [ (value minus population average) squared] divided by population number

Range

the difference between the highest and lowest scores in a distribution

Interquartile range

the difference between the scores that mark the middle 50% of a distribution A Symmetrical Distribution Showing the Interquartile Range as the Difference Between the Scores at the 75th and 25th Percentiles

degrees of freedom

the number of scores that can vary in the calculation of a statistic

Variance

the squared standard deviation of a distribution

Calculate the range for each of these distributions. Was your guess in 5.1 about which has higher variability correct?

3 and 6. Yes, the second set of numbers has a wider range.

Formula for Calculating the Standard Deviation Using Excel

=STDEV(A1:A12) The STDEV function is for samples and will use the formula with the degrees of freedom (n − 1) in its calculation.

interquartile range formula

IQR = Q3 - Q1 X75% - X25%

Consider the two possible sets of data below, both containing 1 to 7 ratings on satisfaction with the food court options at your school. Just by looking at the data themselves, which distribution do you think has higher variability? Why do you think that? 5, 4, 5, 3, 4, 5, 4, 4, 3, 2, 4, 5, 4, 4, 5, 3, 5, 4, 5 2, 4, 1, 3, 5, 6, 7, 3, 4, 5, 6, 4, 5, 6, 1, 2, 7, 3, 4

The second set looks like it has more variability because there are a couple 1s and 7s in the set, whereas the first list does not have any of those extremes.

How are the range and standard deviation used as measures of variability?

The range measures the difference between the highest and lowest scores in a distribution to provide you with a sense of how much of the measurement scale the scores cover. The standard deviation conceptually measures the average distance between the scores and the mean of the distribution. This measure provides you with a sense of how much the scores differ from the center of the distribution.

Discrete variables

measures with whole number scores that cannot be subdivided into smaller units

Calculate the standard deviation for the two sets of data (from populations) below. In what way are the data sets related? How does this relationship influence the standard deviation for the data? 1, 3, 2, 5, 1, 3, 4, 5, 1 2, 6, 4, 10, 2, 6, 8, 10, 2

1.55, 3.01 The range is narrower in the first set (4), so the standard deviation of those nine data points is small. The range of the second set is 8, so we have more variability and the standard deviation is greater.

For the sets of data above, now assume they are data from samples (instead of from populations). How will the standard deviations of the distributions change from those in 5.4?

1.64, 3.28 The standard deviations will be a little larger because we're using degrees of freedom to correct for fewer scores than we would have with a whole population. We're dividing by a number that is smaller than N.

What can we learn about a distribution from measures of variability?

Measures of variability provide some information about how much the scores in a distribution differ from one another.

Range formula

Range = X(max) - X(min)

Steps to get the standard deviation

Step 1: Calculate the deviation between each score and the mean and square the difference. Step 2: Add up the squared deviations to get the sum of squares (SS). Step 3: Divide by N for a population (σ^2) or n − 1 for a sample (s^2) to get the variance. Step 4: Take the square root of the variance to get the standard deviation (σ for a population and s for a sample).

5.3 How do the range and standard deviation compare for different distributions?

The range only uses the two most extreme scores in the distribution, so it is a fairly imprecise measure of variability. It does not provide any information about the scores between these two extremes. The standard deviation uses all the scores in its calculation, so you are getting a more precise measure of variability from the standard deviation.

In your own words, explain why the standard deviation provides a more precise measure of variability for a distribution. In what situation is it the most appropriate measure to use?

The standard deviation gives you an average variance of the data points—how far away from the mean they tend to be. A range merely gives you an idea of lowest and highest boundaries for where a data point might fall. The standard deviation is best used when the mean is your measure of central tendency.

Why does the standard deviation calculation differ for samples and populations?

The standard deviation is the average of the differences between the scores and the mean. For a population, this value is calculated using N to determine the average. However, the sample, as a representative of the population, will have lower variability than the whole population because you are not obtaining scores from every member, only a subset of the population. Thus, we adjust the standard deviation calculation for a sample to account for its lower variability and still provide a good estimate of the variability in the population the sample represents. We do this using n − 1 (which are the degrees of freedom) in our calculation of the average of the deviations between the scores and the mean.

Standard deviation

a measure representing the average difference between the scores and the mean of a distribution