Measures of Central Tendency and Dispersion

Measures of location and central tendency

In statistics, summarizing a complex dataset with a single, representative number is a fundamental task—it helps us identify the distribution’s “center” and understand its typical values. Yet data is different: it can be skewed by outliers, built from rates, or even represent categories instead of numbers.

This diversity is precisely why no single measure of central tendency suffices for all situations. The mean, median, mode, and specialized averages like the geometric or harmonic mean each serve a unique purpose. Choosing the right measure depends on the nature of your data, the presence of extreme values, and what aspect of the distribution you want to highlight.

Let’s imagine we want to compare the performance of two different products based on customer ratings. Simply listing every single rating in a table doesn’t give us a clear overview — it would be much more useful to have a single number that summarizes the overall satisfaction level.
A value like this is called a measure of central tendency, and in statistics, the three most common ones are the mean, median, and mode.
We can say that the mean, median, and mode are measures of central tendency that allow us to summarize a set of data in a single, representative value, depending on what aspect we want to highlight.

More specifically:

The mean is the ratio between the sum of all numerical values and the total number of data points;
The median is the middle value when the data are arranged in order;
The mode is the value that appears most frequently in the dataset.

Arithmetic Mean

The arithmetic mean of a set of numerical data is obtained by adding all the values together and dividing the total by the number of data points.

Example
Consider a courier who must make deliveries across various parts of a city. The journey times for five deliveries on a given day are as follows: 38, 42, 35, 40, and 45 minutes.
To calculate the average journey time per delivery, we sum these values:
38 + 42 + 35 + 40 + 45 = 200
Then we divide by the number of journeys(5):
200 ÷ 5 = 40
So, on average, the courier takes 40 minutes for each delivery trip. This value helps the company to better estimate total delivery times, plan more efficient routes, and provide customers with more accurate delivery windows.

Now that we know what the arithmetic mean is and how to calculate it, we can express it with a formula.

Given numerical data x₁,x₂, …, x_n the arithmetic mean is calculated as (x₁,x₂, …, x_n)/n.

Median

Once we have collected a set of numerical data, we arrange it in ascending or descending order (the order doesn’t matter). The median is the value that lies in the middle position.

However, we must pay attention to two cases:

If the dataset contains an odd number of values, the median is simply the central one.
If it contains an even number of values, there are two central numbers, and the median is the mean of those two values.

Example
A company measures the delivery times (in days) for seven of its recent shipments:
2, 3, 3, 4, 5, 6, 8.
Since there are seven data points (an odd number), the fourth one—4—is the median delivery time.
If the dataset were 2, 3, 3, 4, 5, 6, 8, 9 (eight values), the two central values would be 4 and 5, and the median would be their mean:
(4 + 5) ÷ 2 = 4.5 days.

Mode

The mode of a dataset—also called the modal value—is the value that occurs most frequently (if such a value exists).
A dataset can have one mode, more than one, or none at all if all values occur with the same frequency.

Example
A logistics company analyzes the types of delivery issues encountered in a day. The issues recorded for 12 shipments are as follows:
“Delay”, “Damaged”, “Delay”, “Wrong Address”, “Delay”, “Lost”, “Delay”, “Customer Not Home”, “Delay”, “Customs Hold”, “Delay”, “Damaged”.
To find the mode, we count the frequency of each issue:

Delay: appears 6 times
Damaged: appears 2 times
All other issues appear once.

The value “Delay” occurs most frequently. Therefore, the mode of this dataset is “Delay”.

Variants of the Arithmetic Mean

Weighted Average

A weighted average (or weighted mean) is a variation of the arithmetic mean where each value in the dataset is assigned a specific level of importance, known as a “weight.”

Given n numerical values x₁,x₂,…,x_n and their corresponding weights p₁,p₂,…,p_n, the weighted average is calculated using the following formula:

This can be written more compactly using summation notation :

The purpose of the weighted average is to determine a central value where the original data points do not contribute equally but according to their assigned significance.

Real-World Application and practical example: University Grade Point Average (GPA)

A common real-world application of the weighted average is the calculation of a university student’s final GPA or degree mark.

In this context:

The numerical values (x_i) are the exam scores.
The weights (p_i) are the University Credits (CFU), which define the importance or workload of each course.

The final average is computed by summing the products of each exam score and its corresponding credit value, then dividing by the total number of credits. This ensures that a grade in a more substantial, high-credit course has a greater impact on the final average than a grade in a smaller course.

Scenario: A student has completed four exams with the following scores and credit units (CFUs):

Step 1: Calculate the sum of each score multiplied by its credit value (the numerator).(28⋅9) + (25⋅6) + (30⋅6) + (27⋅9) = 252 + 150 + 180 + 243 = 825

Step 2: Calculate the sum of all the credits (the denominator). 9 + 6 + 6 + 9 = 30

Step 3: Divide the numerator by the denominator to find the weighted average. 825 ÷ 30 = 27.5

Conclusion: The student’s final weighted average is 27.5 out of 30.

Relationship Between Weighted Average and Arithmetic Mean

To understand the algebric and statistical difference between the weighted average and the simple arithmetic mean, we can compare their formulas.

The arithmetic mean of n values x₁,x₂,…,x_n is calculated as:
Arithmetic Mean = (x₁,x₂,…,x_n) / n.
The arithmetic mean is, in fact, a special case of the weighted average where every weight is equal to 1. This means that in the arithmetic mean, all data points are considered equally important.

Geometric Mean

The geometric mean of two or more numerical values is defined as the n-th root of the product of the values, where n is the total number of values. It is a statistical measure that provides specific information about a dataset, particularly useful for understanding multiplicative or proportional relationships.

Given n positive numbers x₁,x₂,…,x_n, the geometric mean is calculated by taking the n-th root of their product. Formula for the geometric mean:

The numbers used to calculate the geometric mean must all be positive. There are two key reasons for this:

If any number were zero, the product would be zero, making the geometric mean zero and rendering the calculation meaningless for most practical purposes.
If n is an even number (like for a square root, fourth root, etc.), the radicand (the value under the root) must be positive, as the even root of a negative number is not a real number.

Using product notation, the formula for the geometric mean of n positive numbers x₁,x₂,…,x_n can be written compactly as:

EXAMPLES

To calculate the geometric mean of 48 and 3:

Multiply the numbers: 48 ⋅ 3 = 144
Take the square root (since there are 2 elements) so the geometric mean is 12.

Relationship with the Arithmetic Mean and Logarithms

An important property of the geometric mean involves logarithms. The logarithm of the geometric mean is equal to the arithmetic mean of the logarithms of the individual values. So log(Geometric mean) = ( log(x₁) + log(x₂)+ ... + log(x_n) ) / n

This relationship is derived from the properties of logarithms and is useful for simplifying calculations, especially with large datasets or in financial models.

Furthermore, for any set of positive numbers, the arithmetic mean is always greater than or equal to the geometric mean.
Arithmetic Mean(x₁,x₂,…,x_n) ≥ Geometric Mean(x₁,x₂,…,x_n)

Harmonic Mean

The harmonic mean is a statistical measure that provides a specific type of average. It is defined as the ratio of the number of values in a dataset to the sum of the reciprocals of those values.

Given n numbers x₁,x₂,…,x_n, the formula is:

Using summation notation, the formula can be written more compactly as:

Example

Let’s calculate the harmonic mean of the numbers 4, 6, and 8.

Step 1: Calculate the sum of the reciprocals (the denominator) 1/4 + 1/6 + 1/8.

The least common denominator is 24, so: 1/4 + 1/6 + 1/8 = 13 / 24.

Step 2: Apply the harmonic mean formula with n=3. Harmonic Mean ≈ 5.538.

Conclusion: The harmonic mean of 4, 6, and 8 is approximately 5.538.

Purpose and Applications of the Harmonic Mean

Unlike more common averages like the arithmetic mean or weighted mean, the harmonic mean is not frequently used in general school studies. Similar to the geometric mean and the quadratic mean, it is a concept typically encountered in specific fields such as Statistics, Finance, and scientific disciplines. Nevertheless, the harmonic mean has many applications in Geometry, Physics, and the aforementioned scientific and financial fields.

Relationship Between Harmonic and Arithmetic Mean

Finally, it’s important to note a key relationship: the harmonic mean is the reciprocal of the arithmetic mean of the reciprocals of the data points. This relationship clearly shows how the harmonic mean “averages” the rates or ratios of the values, rather than the values themselves.

Quadratic Mean (Root Mean Square)

The quadratic mean (also known as the root mean square or RMS) of two or more numbers is defined as the square root of the arithmetic mean of the squares of the numerical values. It is a statistical measure based on the squares of the data points.

Given n numbers x₁,x₂,…,x_n, the formula for the quadratic mean is:

A completely equivalent way to write this formula uses summation notation:

Calculation Example

Let’s imagine we want to calculate the quadratic mean of the numbers 5, 7, 9, and 11.

Step 1: Calculate the sum of the squares of the given values. 5² + 7² + 9² + 11² = 25 + 49 + 81 + 121 = 276

Step 2: Apply the quadratic mean formula with n=4. ≈ 8.3066

Conclusion: The quadratic mean of 5, 7, 9, and 11 is approximately 8.307.

Final Observation

Unless one is studying Statistics, the quadratic mean is rarely used in basic education. In common applications and in middle or high school, it is much more frequent to deal with the arithmetic mean (often simply called “the average”) and the weighted mean.

However, the quadratic mean is crucial in specialized fields such as physics (e.g., for calculating the root mean square speed of molecules) and electrical engineering (e.g., for measuring the effective voltage or current of an AC circuit).

Conclusions

Selecting the right measure hinges on three key factors:

The type of data you are working with (its scale and format).
Whether there are extreme values that could misrepresent the results.
The specific feature of the data’s distribution you wish to emphasize.

This is exactly why no one measure works for every scenario. The mean, median, mode, and other averages like the geometric mean all have distinct roles. Your choice should be informed by your data’s structure, the potential for outliers, and the particular story you want the data to tell.

Measures of dispersion

In statistics, central tendency tells only part of the story. Two datasets can share the same mean or median but have completely different spreads. For instance, two delivery services may both have a 3-day average delivery time, but one is consistently reliable (2-4 days) while the other is unpredictable (same-day to week-long). This difference in variability is captured by measures of dispersion. Measures of variability quantify how closely data points cluster around a central value, such as the mean or median. The critical importance of dispersion lies in its quantification of the spread of data points around the center. This measurement allows for the assessment of consistency, reliability, and risk. A dataset with low dispersion demonstrates high predictability, whereas high dispersion indicates significant variability and uncertainty. In applied settings, such as quality control, a acceptable mean value is insufficient if the dispersion around that mean is excessively high, potentially indicating a process flaw or safety hazard.

The primary measures of dispersion are defined as follows:

The Range is calculated as the difference between the maximum and minimum observed values.
Variance is the mean of the squared deviations of each data point from the dataset’s mean.
The Standard Deviation, being the square root of the variance, returns the measure of spread to the original units of the data.
The Interquartile Range (IQR) measures the statistical spread of the central 50% of the data, providing robustness against the influence of outliers.

Range

The range is a fundamental measure of statistical spread, defined simply as the difference between a dataset’s highest (x_max) and lowest (x_min) values. It is computed using the formula: Range = x_max- x_min. For instance, given the test scores {25, 30, 35, 40, 45}, the range is 45 – 25 = 20.
The range represents the total span covered by the data. A wide range signals high variability among the points, whereas a narrow range indicates that the values are tightly clustered.
The primary advantage of the range is its straightforward calculation and intuitive interpretation. However, it has significant drawbacks: it is highly susceptible to distortion from outliers, as a single extreme value can drastically alter the result. Additionally, it provides no information about the distribution of values between the two extremes. Consequently, the range is best suited for a preliminary, high-level overview of variability—such as in initial quality checks—but is generally considered inadequate for more thorough statistical evaluation.

Variance

Variance is a measure of dispersion that quantifies the average squared distance of each data point from the dataset’s mean. It is calculated differently for a sample (s²) versus a population (σ²):

Sample Variance: s² = Σ(xᵢ – x̄)² / (n – 1)
Population Variance: σ² = Σ(xᵢ – μ)² / N

A higher variance signifies that data points are, on average, farther from the mean, indicating greater spread. A key strength is that it incorporates every value in the dataset, making it fundamental for advanced statistical methods like ANOVA and regression analysis. However, because the calculation involves squaring the differences, the variance is expressed in units that are the square of the original data, which makes its value difficult to relate directly to the original measurements. This interpretability issue is the primary reason the standard deviation is often used in its place. Despite this, variance remains essential in fields such as inferential statistics, financial modeling, and quality control.

Standard Deviation

The standard deviation is defined as the positive square root of the variance. For a variance of 45.5, the standard deviation is √45.5 ≈ 6.75. It is computed directly by taking the square root of the calculated variance. This measure represents the average distance of data points from the mean, and it is expressed in the same units as the original data. This makes its value far more intuitive and meaningful than the variance. It is highly interpretable and the most widely used measure of spread. It is responsive to the fluctuations of all data points around the mean, making it a cornerstone of both descriptive and inferential statistics, especially in fields like finance. Like the variance, it can be disproportionately influenced by outliers. Furthermore, its value is tied to the data’s scale, making direct comparisons between datasets with different units impossible without normalization. It is a fundamental tool in descriptive and inferential statistics, finance, various scientific disciplines, and quality control processes.

Interquartile Range

The IQR is calculated as the difference between the third quartile (Q3) and the first quartile (Q1). Q1 (First Quartile): This is the value that separates the lowest 25% of the data from the rest. In other words, 25% of the data points are less than or equal to Q1. Q3 (Third Quartile): This is the value that separates the lowest 75% of the data from the highest 25%. This means that 75% of the data points are less than or equal to Q3. Subtracting Q1 from Q3 (Q3 - Q1) means calculating the range of the interval that contains the central 50% of the data.
So when data is sorted from lowest to highest, it can be divided into four quarters. Q1 is the boundary between the first and second quarter, while Q3 is the boundary between the third and fourth quarter. The IQR, therefore, measures how “wide” the middle block of data is, excluding the more extreme values found in the two tails (the bottom 25% and the top 25%). This makes it a much more robust measure of spread than the range when outliers are present.
The IQR quantifies the spread of the central half of the dataset, providing a robust measure of variability. It is resistant to outliers, ideal for skewed data and easy to calculate. But disregards the information in the upper and lower tails of the distribution. It reveals fundamental for constructing box plots, detecting outliers, and analyzing asymmetric datasets.

Sources and Further Reading

Statistical Measures of Dispersion: What They Are and How to Calculate Them

Arithmetic Mean, Mode, Median – YouMath

Weighted Average – YouMath

Geometric Mean – YouMath

Harmonic Mean – YouMath

Quadratic Mean – YouMath