# Variability

Variability and its properties

Measures of **variability** describe the amount of variability or spread in the data. The most common measures of
variability are the **range**, the **interquartile range (IQR)**, **variance**, and **standard deviation**.

## Range

The range is a measure of the total spread of values in a quantitative dataset. Unlike other more popular measures of
dispersion, the range actually measures total dispersion, more literally, as **the difference between the largest and
the smallest value in a dataset**.

\( Range = X_{max} - X_{min} \)

Where \(X_{max}\) = Maximum data set value, \(X_{min}\) = Minimum data set value

## Mid-Range

The mid-range of a set of statistical data values is the arithmetic mean of the maximum and minimum values in a data set, defined as:

\( Mid Range = \frac{X_{max} - X_{min}}{2} \)

Where \(X_{max}\) = Maximum data set value, \(X_{min}\) = Minimum data set value

## Variance

Variance measures how far a data set is spread out from their average value. Mathematically, The variance \(σ^2\), is defined as the sum of the squared distances of each term in the distribution from the mean μ, divided by the number of items in the distribution N.

\(Variance = σ^2 = \frac {\Sigma{(X-μ)^2} }{N}\)

Example of samples from two populations with the same mean but different variances. The red population has mean 100 and variance 100 (SD=10) while the blue population has mean 100 and variance 2500 (SD=50).

## Standard Deviation

Standard deviation (σ) is a measure that is used to quantify the amount of variation or dispersion or spread of a data values. σ is a measure of the average distance between the values of the data in the set and the mean.

A low standard deviation indicates that the data points tend to be close to the mean, while a high standard deviation indicates that the data points are spread out over a wider range of values.

Dark blue is one standard deviation on either side of the mean. For the normal distribution, this accounts for 68.27 percent of the set; while two standard deviations from the mean (medium and dark blue) account for 95.45 percent; three standard deviations (light, medium, and dark blue) account for 99.73 percent; and four standard deviations account for 99.994 percent.\( Standard Deviation of population = σ = \sqrt{\frac {\Sigma{(X-μ)^2} }{N}} = \)

\( Standard Deviation of population sample = σ = \sqrt{\frac {\Sigma{(X-μ)^2} }{N-1}} = \)

## Quartiles

Quartiles are the values that divide a list of numbers into quarters.

The

lower quartileis thevalue of the middle of the first setafter dividing the data into two equal sets usingmedian, where 25% of the values are smaller than Q1 and 75% are larger. This first quartile takes the notation Q1.The

upper quartileis the value of the middle of the second setafter dividing the data into two equal sets usingmedian, where 75% of the values are smaller than Q3 and 25% are larger. This third quartile takes the notation Q3. It should be noted that the median takes the notation Q2, the second quartile.

Example 1 – Upper and lower quartiles

Data: 6, 47, 49, 15, 43, 41, 7, 39, 43, 41, 36

Ordered data: 6, 7, 15, 36, 39, 41, 41, 43, 43, 47, 49

Median: 41

Upper quartile = **first quartile Q1** : 43

Lower quartile = **third quartile Q3** : 15

Note: **second quartile Q2** is nothing but the **median**

## Interquartile Range (IQR)

The interquartile range (IQR) is the **distance between the first quartile (Q1) and the third quartile (Q3)**.
50% of the data are within this range.

IQR = Q3 − Q1

## Z-Score

z-score is the number of standard deviations from the mean a data point \( x \) is.

Z-Score = \( \frac { x - μ } {σ} \)

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