Measures of Central Tendency
Measures of central tendency are statistical values that aim to describe the center or typical value of a dataset. The three most common measures are mean, median, and mode.
Mean
The arithmetic mean, often simply called the average, is calculated by summing all values in a dataset and dividing by the number of values. It is the most widely used measure of central tendency.
For a dataset $$x_1, x_2, …, x_n$$, the mean ($$\bar{x}$$) is given by:
$$\bar{x} = \frac{\sum_{i=1}^n x_i}{n}$$
The mean is sensitive to extreme values or outliers, which can significantly affect its value.
Median
The median is the middle value when a dataset is ordered from least to greatest. For an odd number of values, it’s the middle number. For an even number of values, it’s the average of the two middle numbers.
The median is less sensitive to extreme values compared to the mean, making it a better measure of central tendency for skewed distributions[1].
Mode
The mode is the value that appears most frequently in a dataset. A dataset can have one mode (unimodal), two modes (bimodal), or more (multimodal). Some datasets may have no mode if all values occur with equal frequency [1].
Measures of Dispersion
Measures of dispersion describe the spread or variability of a dataset around its central tendency.
Range
The range is the simplest measure of dispersion, calculated as the difference between the largest and smallest values in a dataset [3]. While easy to calculate, it’s sensitive to outliers and doesn’t use all observations in the dataset.
Variance
Variance measures the average squared deviation from the mean. For a sample, it’s calculated as:
$$s^2 = \frac{\sum_{i=1}^n (x_i – \bar{x})^2}{n – 1}$$
Where $$s^2$$ is the sample variance, $$x_i$$ are individual values, $$\bar{x}$$ is the mean, and $$n$$ is the sample size[2].
Standard Deviation
The standard deviation is the square root of the variance. It’s the most commonly used measure of dispersion as it’s in the same units as the original data [3]. For a sample:
$$s = \sqrt{\frac{\sum_{i=1}^n (x_i – \bar{x})^2}{n – 1}}$$
In a normal distribution, approximately 68% of the data falls within one standard deviation of the mean, 95% within two standard deviations, and 99.7% within three standard deviations [3].
Quartiles and Percentiles
Quartiles divide an ordered dataset into four equal parts. The first quartile (Q1) is the 25th percentile, the second quartile (Q2) is the median or 50th percentile, and the third quartile (Q3) is the 75th percentile [4].
The interquartile range (IQR), calculated as Q3 – Q1, is a robust measure of dispersion that describes the middle 50% of the data [3].
Percentiles generalize this concept, dividing the data into 100 equal parts. The pth percentile is the value below which p% of the observations fall [4].
Citations:
[1] https://datatab.net/tutorial/dispersion-parameter
[2] https://www.cuemath.com/data/measures-of-dispersion/
[3] https://pmc.ncbi.nlm.nih.gov/articles/PMC3198538/
[4] http://www.eagri.org/eagri50/STAM101/pdf/lec05.pdf
[5] https://www.youtube.com/watch?v=D_lETWU_RFI
[6] https://www.shiksha.com/online-courses/articles/measures-of-dispersion-range-iqr-variance-standard-deviation/
[7] https://www.khanacademy.org/math/statistics-probability/summarizing-quantitative-data/variance-standard-deviation-population/v/range-variance-and-standard-deviation-as-measures-of-dispersion