What is variability in statistics?

Variability in statistics, also referred to as spread or dispersion, can be described as the extent to which data points in a dataset or distribution differ from the mean and from each other. Measures of variability are often used to describe and compare datasets. Variability is not only useful in statistics but also in finance, psychology among other fields.

Definition of measures of variability?

In descriptive statistics, there are 3 measures of variability; Measures of central tendency, measures of dispersion/variability and measures of frequency. Measures of variability are quantities that are used to describe the mount of spread in a data.

List of Measures of Variation

  • Range – Range shows difference between the smallest/minimum and largest/maximum values in a given dataset. Range is usually calculated by subtracting the smallest number from the largest. For instance, if the largest income is $100,000 and the smallest is $35000 then the range will be $100000-$35000 which is $65000.
  • Interquartile range – Interquartile range is the difference between the first and third quartiles. The formula is IQR=Q3-Q1.
  • Standard deviation – The standard deviation tells you how tightly or apart the data points are to the mean. The standard deviation is usually abbreviated as SD or σ. A small SD indicates that the data is closely knit around the mean while a large SD means that the data points are spread far from the mean.
  • Variance – The variance of a given dataset or variable with a given data sets enables you to know how the data points are spread. A small variance means that the data points are clustered together while a large variance means the data points are more spread apart. In calculation, the variance is the square of the standard deviation.
measures of variability

What does the Interquartile Range tell you?

As the name suggests, interquartile range is very similar to range. The only difference is that in interquartile range interpretation, you are dealing with quartiles and not the whole dataset. Usually there are 3 quartiles: first (Q1), second/middle (Q2) and third quartile (Q3).The first quartile is the middle/central number between the minimum and the median. The second quartile is the median of the data set. The third quartile is the middle value between the median and the maximum values of the data set.

Variability of Grouped and Non-grouped Data

Measures of variability differ across grouped and non-grouped data. Non-grouped data is just a list of values. Non-grouped data usually gives the frequency of a certain variable. Examples of non-grouped data and grouped data include;

variability of grouped and non-grouped data

Measures of Variability for Non-Grouped Data

  • Range –Using the table above, the range can be calculated as 170-114=56
  • Interquartile range is usually calculated after rearranging the non-grouped data in ascending order and then calculating the respective Q1 and Q3.

There are two formulas for calculating the standard deviation for non-grouped data. The sample standard deviation and population formulas are;

sample and population standard deviation formula

Where,

s = sample standard deviation

Ʃ= is the summation sign

ẋ = sample mean

n= number of items/sample size

σ = Population standard deviation

Ʃ= is the summation sign

µ = Population mean

n= number of items/sample size

The relationship between variance and standard deviation is thatvariance is simply the square of the standard deviation after calculation obtained above.

Measures of Variability for Non-grouped Data

The range of grouped data is calculated pretty much the same as that of non-grouped data. Using the table above, the range is 40-0 = 40

The interquartile range formula for grouped data is also the same as that of non-grouped data.

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