Measures of Central Tendency in Statistics
Measures of central tendency are summary values that provide key insights into a statistical distribution.
In statistics, the aim is often to summarize a distribution with a single value that effectively represents the overall trend of the data.
There are various types of central tendency indicators:
- Mean-based measures
These measures take all values in the distribution into account and are generally used to find the central value of the distribution. Examples include the arithmetic mean, geometric mean, harmonic mean, quadratic mean, and others.Example. Consider a set of values: $$ 5 \ , \ 7 \ , \ 2 , \ 12 \ , \ 4 $$ The arithmetic mean gives us a central value around which the other elements of the distribution tend to cluster. $$ \mu = \frac{5+7+2+12+4}{5} = \frac{30}{5} = 6 $$
- Median-based measures
These are calculated using only certain values from the distribution and can reflect non-central trends as well. Examples include the median, quantiles, quartiles, quintiles, deciles, percentiles, and mode.Example. Let’s use the same distribution of values as before: $$ 5 \ , \ 7 \ , \ 2 , \ 12 \ , \ 4 $$ Sorting these values in ascending order gives us: $$ 2 \ , \ 4 \ , \ 5 , \ 7 \ , \ 12 $$ The first quartile is the value that divides the ordered set into two parts: the first containing 1/4 of the data, and the second with 3/4. $$ \underbrace{ 2 } \ , \ 4 \ , \ \underbrace{ 5 , \ 7 \ , \ 12 \ } $$ In this case, the first quartile is 4. $$ Q_1 = 4 $$
- Other measures of central tendency
There are also other indicators that capture specific characteristics of a distribution. Examples include the mode, minimum value, maximum value, and others.Example. Consider this distribution: $$ 3 \ , \ 6 \ , \ 2 \ , \ 8 \ , \ 2 $$ The mode is the value that appears most frequently. $$ 3 \ , \ 6 \ , \ \color{red}2 \ , \ 8 \ , \ \color{red}2 $$ In this case, the mode is 2 because it occurs twice, while all other values appear only once. $$ \mu_o = 2 $$
What are the main measures of central tendency?
The most common central tendency measures are:
- Arithmetic mean
The arithmetic mean is the most common and widely used measure of central tendency. It's straightforward to calculate. - Weighted mean
In this case, each value in the dataset is assigned a weight. There are different types of weighted means (e.g., arithmetic, geometric, etc.). - Geometric mean
The geometric mean is often used to calculate average percentage changes for phenomena that evolve over time. - Harmonic mean
The harmonic mean is particularly useful for averaging values that are rates or ratios, such as in determining the average price of a good to assess purchasing power. - Quadratic mean
The quadratic mean is used when assessing positive or negative deviations from a fixed mean value, or when outliers significantly distant from the center of a data distribution need to be taken into account. - Median
The median is especially valuable because it's unaffected by extreme values in the dataset. - Mode
The mode represents the value that appears most frequently in a dataset. - Quantiles
Quantiles (quartiles, deciles, percentiles) divide an ordered dataset into equal parts, providing thresholds below which a given percentage of the data falls.
Relationship between the means
It can be demonstrated that there’s an ordered relationship between the arithmetic mean (A), geometric mean (G), harmonic mean (M), and quadratic mean (Q):
$$ M < G < A < Q $$
In other words, the harmonic mean (M) is smaller than the geometric mean (G), which is smaller than the arithmetic mean (A), and finally, the quadratic mean (Q) is the largest.
Example
To demonstrate this relationship, let's consider a simple example with two numbers: 4 and 9.
- Arithmetic mean (A)
The arithmetic mean is the sum of the values divided by the number of values: $$ A = \frac{4 + 9}{2} = \frac{13}{2} = 6.5 $$ - Geometric mean (G)
The geometric mean is the n-th root of the product of the values, where n is the number of values: $$ G = \sqrt{4 \times 9} = \sqrt{36} = 6 $$ - Harmonic mean (M)
The harmonic mean is calculated as the reciprocal of the mean of the reciprocals: $$ M = \frac{2}{\frac{1}{4} + \frac{1}{9}} = \frac{2}{\frac{9 + 4}{36}} = \frac{2 \times 36}{13} \approx 5.54 $$ - Quadratic mean (Q)
The quadratic mean is the square root of the mean of the squares of the values: $$ Q = \sqrt{\frac{4^2 + 9^2}{2}} = \sqrt{\frac{16 + 81}{2}} = \sqrt{\frac{97}{2}} = \sqrt{48.5} \approx 6.96 $$
As we can see from the calculations, the order is maintained:
$$ M \approx 5.54 < G = 6 < A = 6.5 < Q \approx 6.96 $$
This confirms that the harmonic mean is the lowest, followed by the geometric mean, the arithmetic mean, and finally the quadratic mean.
And so on.
