Measures of Variability

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I recommend that you learn about the mean, median, and mode before reading this post.

Range:

The range is the difference between the highest and lowest values in the set of data.

Example:

\displaystyle Data = \{1, 2, 3, 4, 5\} \\ Range = (5 - 1 ) = 4

Interquartile Range:

The interquartile range is very similar to the range, but instead of using the highest and lowest values, we use a value that is 25% lower than the median and 25% higher than the median. This allows us to find the range of the central 50% of the data elements.

Before we can find the interquartile range, we need to find the first and third quartiles. These are the values that are 25% higher and lower than the median.

\displaystyle Data = \{0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10\}

Example - First Quartile:

First we take the total number of data elements, add one, and divide by four to get three.

The value that we just calculated means that the third value in the set of data is the first quartile. In this case the first quartile is two.

Example - Third Quartile:

Because we’ve already calculated the value three when finding the first quartile, we just need to take the third last value in the data set as the third quartile.

In this case the third quartile is eight.

Example - Inter-Quartile Range (IQR):

\displaystyle IQR = (Third Quartile - First Quartile) = 8 - 2 = 6

This may seem like a lot to remember and do at first, but it’s very easy to do and remember if you do one or two practise problems a day for a few days.

Variance:

The variance is somewhat like an average of how far away every element in the data set is from the mean. The variance is often denoted by σ.

When the elements of the data set are close to the mean, then the variance is small. When the elements are far from the mean, then the variance is large.

One thing to note is that when it comes to class work, you will generally use the standard deviation instead of the variance.

Example - Variance of Population:

\displaystyle Data = \{1, 2, 3\} \\ Mean = 2 \\ X = (1 - 2)^2 + (2 - 2)^2 + (3 - 2)^2 \\ \sigma^2 = \frac{X}{3} = \frac{2}{3}

Example - Variance of Sample:

\displaystyle Data = \{1, 2, 3\} \\ Mean = 2 \\ X = (1 - 2)^2 + (2 - 2)^2 + (3 - 2)^2 \\ \sigma^2 = \frac{X}{3 - 1} = \frac{2}{2} = 1

As you can see in the two examples above, the only difference between taking the variance of a population and of a sample is that, with the sample, you need to subtract one from the denominator.

Standard Deviation:

The standard deviation also shows the average of how far away every element in the data set is from the mean, but it is often used more on homework and in class than the variance is. The standard deviation is often denoted by σ.

Assuming the variance has already been found, the standard deviation is the square root of the variance.

Example - Standard Deviation of Population:

\displaystyle Variance = 2/3 \\ \sigma = sqrt(\frac{2}{3}) = 0.81649658092

Example - Standard Deviation of Sample:

\displaystyle Variance = 1 \\ \sigma = \sqrt1 = 1