Introduction

The central tendency is the point of accumulation of observations near the centres of any given data. The three types of central tendencies are Mean, Mode, and Median. How the data varies is measured as the variation of observations from these central tendencies and there are several ways in which we can perform this measurement. The most common ones are Range, Standard Deviation, and Variance. However, in this article, we will focus on the simplest measure of variation, which is Range. We will also cover the three central tendencies, Mean, Mode, and Median.

Central tendencies

Arithmetic Mean (Average)

The arithmetic mean, also known as the mean, is the average value of a set of data obtained from an experiment or survey. It is calculated by dividing the sum of all the observations by the total number of observations. There are two types of arithmetic means:

1. Simple Arithmetic Mean: Where each observation contributes equally to the data.

2. Weighted Arithmetic Mean: Where certain observations contribute more to the data than others.

It must be noted that weighted arithmetic mean differs from the arithmetic mean of grouped data and the two must not be confused.

Arithmetic Mean Formula

Let, represent the n number of observations of a given experiment. The Arithmetic mean of these is given by,

Now suppose some of the observations are being repeated in the data set and their frequencies of occurrence are given by f1, f2, f3 and so on. In that case, we can write the arithmetic mean as:

Here, N represents the total number of observations and is given as N = f1 + f2 +  f3 

When data is grouped, we can calculate its mean in a number of ways, chief among them being the following:

1. Direct method.

2. Assumed mean method.

3. Step Deviation method.

Geometric Mean

Geometric mean is another type of central tendency, similar to Arithmetic mean. It is calculated by taking the nth root of the product of all the observations, where n is the total number of observations in the data. The formula for the geometric mean is given by:

Range

In general, the range refers to the collection of all the values that a variable can take.

Arithmetic Range

Range in statistics is a measure of variability that quantifies the spread of data from the central tendencies. It is calculated as the difference between the highest and lowest observations in a dataset. The range represents the maximum possible variation between any two observations in the data, and no two observations can have a difference greater than the arithmetic range of the dataset.

Range Formula

To find the range, we can just subtract the smallest value from the highest one that the variable is taking. Here is an example: there are ten students in a class who were given a quiz and the marks they scored are listed as follows:

3, 6, 6, 8, 10, 9, 4, 2, 7, 5

Then the range is given by the highest minus the lowest value. 

Range = Highest – lowest

Range = 10 – 2

Range = 8

Median

The median value of a data set represents the center-most data in the set. To find the median, the data must be placed in either ascending or descending order and without doing so, we cannot find the media. Two cases arise when we try to find the median:

1. The number of observations, n, is odd. In such a case, the median of the data is the value of the

 

2. The number of observations is even. In this case, the median value is given by the arithmetic mean of the

 

Mode

To determine the most frequently occurring observation in a dataset, we use another central tendency called Mode. We can find the mode by tabulating the frequency of each observation and identifying the one with the highest frequency. 

An interesting point to note is that the mode is not a true central tendency, as the most repeated observation in a dataset can occur anywhere. In real-life experiments, however, the most frequent observation often occurs near the center of the data, which is why mode can be used as a central tendency for real-life-based surveys and data.

Summary

Central tendencies are measures of the most common or average observation of data. The three types of central tendencies are mean (average), median (middle observation), and mode (most common observation). Here is a summary:

• Mean: The average value of the data set.

• Median: The centre-most data of the set.

• Mode: The most frequently occurring data in the set.

• Geometric mean: The nth root of the product of all observations.

Frequently Asked Questions 

1. State the relationship between Arithmetic and Geometric Mean of a given set of numbers.

This relation is also known as the AM-GM inequality, and it states that the Arithmetic mean of two or more non-negative numbers is always greater than or equal to their geometric mean. Thus,

2. What is the relationship between Arithmetic, Geometric and Harmonic mean of two numbers?

Given two numbers a and b, we have:

3. What is the relationship between the three central tendencies?

This relation is known as the empirical relation and it is stated as follows:

4. Determine the Range and the three Central Tendencies for the following given data

24, 11, 6, 9, 10, 5, 3, 19, 21, 8, 27, 23, 25, 15, 17, 14, 19, 28, 30, 14, 22, 9, 14, 15, 13.

We first arrange the data in ascending order.

3, 5, 6, 8, 9, 9, 10, 11, 13, 14, 14, 14, 15, 15, 17, 19, 19, 21, 22, 23, 24, 25, 27, 28, 30.

There are 25 observations in total with the highest one being 30 and the lowest one being 3. Thus, the range is 

Since there are 25 observations in total, the 13th observation is the median, which is 15 and the mode is the most occurring data, which is 14.

5. Verify the Empirical relationship for the previous question.

We have:

Mean = 16.04

Median = 15

Mode = 14

It is worth noting that the empirical relation holds best for large data sets.