Mean Median Mode – Difference, Formula, and Examples

In statistics, the mean, median, and mode are the three measures of central tendency. While describing a data set, we identify the central position of the data. This is known as a measure of central tendency. We come across data every day. Newspapers, articles, bank statements, mobile and electricity bills contain them. It is impossible to list them all; they are everywhere. Considering only a few representatives of the data, we might be able to find out some important features of the data. This can be achieved by using measures of central tendency or averages, namely mean, median, and mode.

In the following sections, we will examine mean, median, and mode using examples.

Mean, Median and Mode in Statistics

The measures of central tendency, mean, median, and mode, are used to analyze the various characteristics of a particular set of data. A central tendency measure describes a set of data by identifying its central position in the set as a single value. It can be thought of as a tendency for data to cluster around a middle value. The three most common measures of central tendencies in statistics are Mean, Median, and Mode. Depending on what kind of data we have, we can choose the best central tendency measure.

Let’s start by understanding the meaning of each of these terms.

Mean

The arithmetic mean of a given dataset is the sum of all observations divided by the number of observations. A cricketer’s scores in five ODI matches, for example, are 12, 34, 45, 50, 24. In order to calculate his average score in a match, we use the mean formula to calculate the arithmetic mean of the data:

Mean = Sum of all observations/Number of observations

Mean = (12 + 34 + 45 + 50 + 24)/5

Mean = 165/5 = 33

Mean is denoted by x̄ (pronounced as x bar).

Types of Data

Data can be present in raw form or tabular form. Let’s find the mean in both cases.

Raw Data

Let x1x1, x2x2, x3x3 ……xnxn be n observations.

We can find the arithmetic mean using the mean formula.

Mean, x̄ = x1+x2+…xnnx1+x2+…xnn

Example: If the heights of 5 people are 142 cm, 150 cm, 149 cm, 156 cm, and 153 cm.

Find the mean height.

Mean height, x̄ = (142 + 150 + 149 + 156 + 153)/5
= 750/5
= 150

Mean, x̄ =150 cm

Thus, the mean height is 150 cm.

Median

The value of the middlemost observation, obtained after arranging the data in ascending order, is called the median of the data.

For example, consider the data: 4, 4, 6, 3, 2. Let’s arrange this data in ascending order: 2, 3, 4, 4, 6. There are 5 observations. Thus, median = middle value i.e. 4. We can see here: 2, 3, 4, 4 , 6 (Thus, 4 is the median)

Case 1: Ungrouped Data

• Step 1: Arrange the data in ascending or descending order.
• Step 2: Let the total number of observations be n.

To find the median, we need to consider if n is even or odd. If n is odd, then use the formula:

Median = (n + 1)/2^th observation

Example 1: Let’s consider the data: 56, 67, 54, 34, 78, 43, 23. What is the median?

Solution:

Arranging in ascending order, we get: 23, 34, 43, 54, 56, 67, 78. Here, n (no.of observations) = 7

So, (7 + 1)/2 = 4

∴ Median = 4^th observation

Median = 54

If n is even, then use the formula:

Median = [(n/2)^th obs.+ ((n/2) + 1)^th obs.]/2

Mode

The value which appears most often in the given data i.e. the observation with the highest frequency is called a mode of data.

Case 1: Ungrouped Data

For ungrouped data, we just need to identify the observation which occurs maximum times.

Mode = Observation with maximum frequency

For example in the data: 6, 8, 9, 3, 4, 6, 7, 6, 3 the value 6 appears the most number of times. Thus, mode = 6. An easy way to remember mode is: Most Often Data Entered. Note: A data may have no mode, 1 mode, or more than 1 mode. Depending upon the number of modes the data has, it can be called unimodal, bimodal, trimodal, or multimodal.

The example discussed above has only 1 mode, so it is unimodal.

Case 2: Grouped Data

When the data is continuous, the mode can be found using the following steps:

• Step 1: Find modal class i.e. the class with maximum frequency.
• Step 2: Find mode using the following formula:

Mode = l+[fm−f12fm−f1−f2]×hl+[fm−f12fm−f1−f2]×h

where,

• l = lower limit of modal class,
• fm=fm= frequency of modal class,
• f1=f1= frequency of class preceding modal class,
• f2=f2= frequency of class succeeding modal class,
• h = class width

Mean, Median, and Mode Formulas

We covered the formulas and methods to find the mean, median, and mode for grouped and ungrouped sets of data. Let us summarize and recall them using the list of mean, median, and mode formulas given below,

Mean formula for ungrouped data: Sum of all observations/Number of observations

Mean formula for grouped data: x̄ = x1f1+x2f2+….xnfn f+f2+…..fnx1f1+x2f2+….xnfn f+f2+…..fn

The median formula for ungrouped data: If n is odd, then use the formula: Median = (n + 1)/2^th observation. If n is even, then use the formula: Median = [(n/2)^th obs.+ ((n/2) + 1)^th obs.]/2

Median formula for grouped data: Median = l+[n2−cf]×hl+[n2−cf]×h

where,

• l = lower limit of median class
• c = cumulative frequency of the class preceding the median class
• f = frequency of the median class
• h = class size

Mode formula for ungrouped data: Mode = Observation with maximum frequency

Mode formula for grouped data: Mode = l+[fm−f12fm−f1−f2]×hl+[fm−f12fm−f1−f2]×h

where,

• l = lower limit of modal class,
• fmfm = frequency of modal class,
• f1f1 = frequency of class preceding modal class,
• f2f2 = frequency of class succeeding modal class,
• h = class width

Relation Between Mean, Median and Mode

The three measures of central values i.e. mean, median, and mode are closely connected by the following relations (called an empirical relationship).

2Mean + Mode = 3Median

For instance, if we are asked to calculate the mean, median, and mode of continuous grouped data, then we can calculate mean and median using the formulas as discussed in the previous sections and then find mode using the empirical relation.

For example, we have data whose mode = 65 and median = 61.6.

Then, we can find the mean using the above mean, median, and mode relation.

2Mean + Mode = 3 Median

∴2Mean = 3 × 61.6 – 65

∴2Mean = 119.8

⇒ Mean = 119.8/2

⇒ Mean = 59.9

Difference Between Mean and Average

In everyday life, the term average is often used to denote a value that is typical for a group of quantities. An example would be the average rainfall in a month or the average age of employees in an organization. In an article we might read, “People spend an average of two hours a day on social media. ” By saying average, we understand that not everyone spends that much time on social media. However, some people spend more time and some spend less time.

Nevertheless, we can infer from the term average that 2 hours per day is a good indicator of how much time is spent on social media. Despite the fact that average and mean are not the same, most people use them interchangeably.

• The average is the value that indicates what is most likely to be expected.
• This is a useful way to summarize large amounts of data.

Averages tend to be centered with the observations arranged in ascending order of magnitude. Therefore, we call an average a measure of the central tendency of the data. Averages are of different types. An average is what we call the mean, i.e. the arithmetic mean. Median and mode are positional averages, while mean is the mathematical average.

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