Mean, Median, and Mode: Formulas, Definitions, and Solved Examples

Mean, Median, Mode Formula: We live in an era of big data, where information is present in every facet of life, from news cycles to smartphone usage. To derive meaningful insights from large datasets, we rely on specific representatives of the data known as central tendency or averages: mean, median, and mode. When interpreting a dataset, we identify a central position, which is known as the measure of central tendency. Statistics primarily uses 3 measures of central tendency: Mean, Median, and Mode. In this article, we break down these concepts, provide clear definitions, and offer solved examples to help you master the mean, median, and mode formulas.  

Mean Median Mode Formula

The mean represents the arithmetic average of a dataset. The median is the middle value when your data is arranged in ascending order. The mode is the most frequently occurring observation within the set. Below, we explain the specific formulas for calculating each. 

Mean Formula

Mean is defined as the arithmetic average of a dataset. It is calculated by finding the sum of all observations divided by the total count of observations. For instance, consider a batsman scoring 35, 24, 49, 57, and 16 runs in five T20 matches. To find his average, we apply the standard mean formula: 

Mean = Sum of all observations / Total number of observations

Mean or average runs for the batsman = (35+24+49+57+16) / 5

Mean = 181 / 5 = 36.2

In statistics, the Mean is symbolized by x̄ (pronounced as ‘x-bar’).

Here,

Σ denotes the summation or addition

X denotes the individual observations

N denotes the total count of observations.

For tabular or frequency data, use the following mean formula:

Mean = Σfx / Σf

Where Σf = N

The calculated mean is not necessarily a number from the original dataset. It provides an arithmetic average, which means the resulting figure may not be identical to any single data point provided.

Mean Formula Solved Questions

Data can be presented as a raw list or in a table. Below are solved problems for both raw data and grouped tabular data. 

Raw Data Problems

Suppose x1, x2, x3, . . . , xn are the observations.

We calculate the mean using the formula below:

Mean x̄ = (x1 + x2 + ... + xn) / n

Question 1: If 5 boys have heights of 132 cm, 140 cm, 143 cm, 148 cm, and 151 cm, what is the mean height?

Solution: Mean Height x̄ = (132 + 140 + 143 + 148 + 151) / 5

= 714 / 5 = 142.8

Hence, the mean height x̄ of the five boys is 142.8 cm.

Question 2: The mean of three numbers is 12. If one number is excluded, the mean becomes 8. Find the excluded number.

Solution: The total number of observations (n) = 3

The mean value x̄ = 12

By using the Mean Formula, x̄ = Σx / n

Σx = x̄ * n = 12 * 3 = 36

The sum of the 3 observations is 36.

Let the excluded number be z.

The sum of the remaining two observations = 36 – z

The mean of the 2 remaining observations = (36 – z) / 2

8 = (36 – z) / 2

36 – z = 16

z = 36 - 16 = 20

Hence, the excluded number is 20.

Tabular Data (Frequency Distribution) Problems

When data is presented in a frequency table, apply the following Mean formula:

Mean x̄ = (x1f1 + x2f2 + ……+ xnfn) / (f1 + f2 + ... + fn)

Question 1: Calculate the mean value of the following distribution:

x	2	4	7	11	17
f	4	12	11	5	3

Solution: Arrange the tabular data as follows to find the mean value:

xᵢ	fᵢ	xᵢfᵢ
2	4	8
4	12	48
7	11	77
11	5	55
17	3	51
	∑ fᵢ = 35	∑ xᵢ fᵢ = 239

Mean x̄ = (Σxi fi) / (Σfi)

x̄ = 239 / 35 = 6.82 (Approx.)

Hence, the mean value of the distribution is approximately 6.82.

Question 2: A table displays the number of students visiting a library. Calculate the average daily visits.

Number of Students	Number of days visiting Library
0-10	3
10-20	5
20-30	8
30-40	10
40-50	6
50-60	7

Solution: In this case, we use the classmark (mid-point of a class) concept. Let x1, x2, . . . , xn be the class marks for the categories. This yields the following table:

Classmark (xᵢ)	Frequency (fᵢ)	xᵢfᵢ
5	3	15
15	5	75
25	8	200
35	10	350
45	6	270
55	7	385
Total	∑ fᵢ = 39	∑ fᵢxᵢ = 1295

Mean x̄ = (Σ xifi) / (Σ fi)

x̄ = 1295 / 39 = 33.2 = 33 (Approx.)

Hence, the average number of students visiting the library per day is 33.

Median Formula

The median is the middle value in a sorted dataset, whether arranged in ascending or descending order. Therefore, the median refers to the middlemost observation. For example, in the dataset 3, 7, 15, 2, 6, 6, we first arrange them: 2, 3, 6, 6, 7, 15. Since there are 6 observations, the median is the average of the middle two values. 

When the total number of observations (n) is odd, use the following median formula:

When the total number of observations (n) is even, use the following median formula:

Median Formula Solved Questions

Data can be given in ungrouped or grouped patterns. Here are solved problems for both ungrouped and grouped data cases. 

Ungrouped Data Problems

For ungrouped data, always sort the numbers in ascending or descending order first, then determine if the number of observations (n) is odd or even to choose the correct formula. 

Question 1: What is the median of the following dataset: 61, 46, 49, 64, 57, 38, 21?

Solution: Since the data is ungrouped, we arrange it in ascending order. 

Sorted data: 21, 38, 46, 49, 57, 61, 64

The total number of observations (n) = 7 (odd)

Apply the median formula for an odd number of observations:

Median = (n+1) / 2 = (7 + 1) / 2 = 4th observation

The median of the dataset is 49.

Question 2: Find the median for the data: 5, 7, 2, 4, 8, 4.

Solution: First, sort the data in ascending order. 

Sorted data: 2, 4, 4, 5, 7, 8

The total number of observations (n) = 6 (even)

Apply the median formula for an even number of observations:

Median = [(n / 2)th observation + ((n / 2) + 1)th observation] / 2

Median = (3rd observation + 4th observation) / 2

Median = (4 + 5) / 2 = 9 / 2 = 4.5

Grouped Data Problems

For continuous grouped frequency distributions, we use a specific median formula. The median class is the interval where (n / 2) falls.

Here, l = Lower limit of the median class

c = Cumulative frequency of the class preceding the median class

f = Frequency of the median class

h = Class size

n = Total number of observations (Σ fi) 

Question 1: Calculate the median marks for the given distribution:

Classes	0-10	10-20	20-30	30-40	40-50
Number of Students	10	2	12	18	3

Solution: Since the data is continuous and grouped, we calculate cumulative frequencies first.

Classes	Number of students	Cumulative frequency
0-10	10	10
10-20	2	2 + 10 = 12
20-30	12	12 + 12 = 24
30-40	18	18 + 24 = 42
40-50	3	3 + 42 = 45

Total observations (n) = Σ fi = 45

n/2 = 45/2 = 22.5 

The Median class containing 22.5 is 20-30.

Parameters: l = 20, f = 12, c = 8, h = 10

Applying the median formula:

Median = l + [((n / 2) − c) / f] × h

Median = 20 + [(22.5 - 8) / 12] × 10

Median ≈ 32.08

Question 2: Calculate the median value for the given distribution:

Category	0-10	10-20	20-30	30-40	40-50
Frequency	7	22	9	28	13

Solution: Using cumulative frequencies for grouped data.

Category	Frequency	Cumulative Frequency
0-10	7	7
10-20	22	22 + 7 = 29
20-30	9	9 + 29 = 38
30-40	28	28 + 38 = 66
40-50	13	13 + 66 = 79

Total observations (n) = Σ fi = 79

n/2 = 79 / 2 = 39.5 

The Median class containing 39.5 is 30 - 40.

Parameters: l = 30, f = 28, c = 38, h = 10

Applying the median formula:

Median = l + [((n / 2) − c) / f] × h

Median = 30 + [(39.5 - 38) / 28] × 10

Median ≈ 30.53

Mode Formula

The mode is the value that appears most frequently in a dataset. Put simply, the observations with the highest frequency represent the mode. The range denotes the difference between the highest and lowest values. Remember, MODE stands for "Most Often Data Entered." A dataset may have no mode, one mode, or multiple modes; datasets with these patterns are known as unimodal, bimodal, trimodal, or multimodal.

Here,

L = Lower limit of the modal class.

h = Size of the class interval.

fm = Frequency of the modal class.

f1 = Frequency of the class preceding the modal class.

f2 = Frequency of the class succeeding the modal class.

Mode Formula Solved Questions

Data can be ungrouped or grouped. Here are solved problems for both. For ungrouped data, you simply identify the value that repeats most often.

Mode = Observation having the highest frequency

Question 1: Given student ages {17,14,15,17,16,18,17}, find the mode.

Solution: Dataset: {17,14,15,17,16,18,17}

Using the mode formula:

Mode = Observation with the highest frequency

Since one number appears more frequently than others, this is unimodal data.

The value of the mode is 17.

Question 2: Calculate the mean, median, mode, and range for the list: 12, 17, 14, 15, 12, 17, 12, 20, 12

Solution: Dataset: 12, 17, 14, 15, 12, 17, 12, 20, 12

To find the mean:

Mean = (12 + 17 + 14 + 15 + 12 + 17 + 12 + 20 + 12) / 9 = 131 / 9 = 14.55

To find the median, first sort the list in ascending order:

Sorted data: 12, 12, 12, 12, 14, 15, 17, 17, 20

There are 9 observations (odd). The median is at (9 + 1) / 2 = 5th position.

Thus, the median is 14.

To find the mode, identify the value that occurs most often.

12 is the mode, as it repeats 4 times.

To find the range, subtract the smallest value from the largest value (20 - 12).

The range is 8.

Mean = 14.55

Median = 14

Mode = 12

Range = 8