14th February 2024

Describing data

Summary

A dataset can be summarised by describing its central tendency and variability.

Measures of central tendency

There are three types of measures of central tendency – mean, median, and mode.

Mean is the average of all of the data. It is obtained by adding all the values and dividing the observations. 

Where ∑ 𝑥=sum of all values, n=number of observations

The median is the middle point of all observations. The median is where half of the observations lie below and the other half lie above. 

Mode is the most commonly recorded observation. 

The central tendency is best described using the mean for normally distributed data. Interestingly, for normally distributed data, as data is spread equally on both sides of the mean, mean = median = mode.

The central tendency is best described using the median for non-normally distributed data, as outliers do not influence it. Other characteristics of measures of central tendencies in non-normally distributed data include:

• Distribution with positive skew: mode < median < mean 
• Distribution with negative skew: mean < median < mode

Measures of variability

Measures of variability describe the spread of observations. The type of measures of variability that are used depends on the data distribution.

The simplest way of describing the spread of data is by its range. The range of data states the lowest (min) and highest (max) data points. However, it provides no other information regarding the data’s distribution.

When data is normally distributed, data is symmetrically spread around the mean. Variance and standard deviation are used to describe variability.

When the data is not normally distributed, it is better to use the interquartile range. Interquartile range is the difference between observations at the 25th percentile (a.k.a. first quartile) and 75th percentile (a.k.a. third quartile).

When we repeatedly sample a population randomly, we obtain different datasets using different means. How do we measure the variability of these means? We use the standard error of the mean.

References

Frost, J. (2018a, February 12). Measures of Central Tendency: Mean, Median, and Mode. Statistics By Jim. http://statisticsbyjim.com/basics/measures-central-tendency-mean-median-mode/ 

Frost, J. (2018b, March 2). Measures of Variability: Range, Interquartile Range, Variance, and Standard Deviation. Statistics By Jim. http://statisticsbyjim.com/basics/variability-range-interquartile-variance-standard-deviation/

Frost, J. (2021, May 24). Standard Error of the Mean (SEM). Statistics By Jim. https://statisticsbyjim.com/hypothesis-testing/standard-error-mean/