Section 5.2: Measures of Variation

Section 5.2: Measures of Variation

Another component of describing a data set is how much “Spread” there is in the data set. In other words, how much the data in the distribution vary from one another. It may seem like once we know the center of a data set, we know everything there is to know. The first example will demonstrate why we need measures of variation (or spread).

There are several ways to measure this “Spread” of the data. The three most common measures are the range, standard deviation, and quartiles. In this section we will learn about the range and standard deviation. We will discuss quartiles in the following section.

We will focus first on the simplest measure of spread, called the range.

Range

The range is the difference between the maximum value and the minimum value of the data set.

Standard Deviation

The standard deviation is a measure of variation based on measuring how far, on average, each data value deviates, or is different, from the mean. A few important characteristics:

1) Standard deviation is always positive. Standard deviation will be zero if all the data valuesare equal, and will get larger as the data spreads out.
2) Standard deviation has the same units as the original data.
3) Standard deviation, like the mean, can be highly influenced by outliers.

Using the data from Section D: 0 5 5 5 5 5 5 5 5 10, we could compute for each data value the difference between the data value and the mean. This will give us an idea of “how far” each value in the data set lies away from the mean.

We would like to get an idea of the “average” deviation from the mean, but if we find the average of the values in the second column the negative and positive values cancel each other out (this always happens), so instead we square every value in the second column:

We then add the squared deviations up to get 25 + 0 + 0 + 0 + 0 + 0 + 0 + 0 + 0 + 25 = 50. Ordinarily we would then divide by the number of scores, n, (in this case, 10) to find the mean of the
deviations. But we only do this if the data set represents a population; if the data set represents a sample (as it almost always does), we instead divide by n – 1 (in this case, 10 ─ 1 = 9).

So in our example, we would have [latex]\frac{50}{10}[/latex] = 5 if section D represents a population and [latex]\frac{50}{9}[/latex] = about 5.56 if section D represents a sample. These values (5 and 5.56) are called, respectively, the population variance and the sample variance for section D.

Variance can be a useful statistical concept, but note that the units of variance in this instance would be points – squared since we squared all of the deviations. What are points – squared? Good question. We would rather deal with the units we started with (points in this case), so to convert back we take the square root and get:

Population standard deviation: [latex]\sqrt\frac{50}{10}[/latex] ≈ 2.2

Sample standard deviation: [latex]\sqrt\frac{50}{9}[/latex] ≈ 2.4

If we are unsure whether the data set is a sample or a population, we will usually assume it is a sample, and we will round answers to one more decimal place than the original data, as we have done above.