Mean, Median and Mode

Mean, median, and mode are measures of central tendency that help describe the characteristics of a data set. They provide useful insights into the data by identifying its average value (mean), middle value (median), and most frequent value (mode).

These measures are used in all kinds of real-life situations, such as finding the average salary of employees, determining the median age of a class, or identifying the most popular sport played in a club.

Mean

Mean is the sum of all the values in the data set divided by the number of values in the data set. It is also called the Arithmetic Average. The Mean is denoted as x̅.

Formula

The formula to calculate the mean is Mean\bar{x} = \frac{\sum x_i}{n}

If x₁,x₂, x₃, ..., x_n are the values of a data set, then the mean is calculated as:

x̅ =  (x₁ + x₂ + x₃ + . . . + x_n) / n

ForExample: Find the mean of data sets 10, 30, 40, 20, and 50.

Solution:

Mean of the data 10, 30, 40, 20, 50 is
Mean = (10 + 30 + 40 + 20+ 50) / 5 = (150)/5 = 30

Median

A median is a middle value for sorted data. The sorting of the data can be done either in ascending order or descending order. A median divides the data into two halves. 

Formula

1) If the number of values (n value) in the data set is even, then the formula to calculate the median is:

2) If the number of values (n value) in the data set is odd, then the formula to calculate the median is:

Example: Find the median of the given data set 30, 40, 10, 20, and 50.

Solution:

Step 1: Order the given data in ascending order as:
10, 20, 30, 40, 50

Step 2: Check n (number of terms of data set) is even or odd and find the median of the data with respective 'n' value.

Step 3: Here, n = 5 (odd)

Median = [(n + 1)/2]^th term
Median = [(5 + 1)/2]^th term
= 30

Mode

A mode is the most frequent value or item of the data set. A data set can generally have one or more than one mode value. If the data set has one mode, then it is called "unimodal." Similarly, if the data set contains 2 modes, then it is called "Bimodal" and if the data set contains 3 modes, then it is known as "Trimodal".

If the data set consists of more than one mode, then it is known as "multi-modal" (can be bimodal or trimodal). There is no mode for a data set if every number appears only once.

The formula to calculate the mode is:

Example: Find the mode of the given data set: 1, 2, 2, 2, 3, 3, 4, 5.

Solution:

Given set is {1, 2, 2, 2, 3, 3, 4, 5}

As the above data set is arranged in ascending order.

By observing the above data set we can say that,

Using the formula
Mode = Highest Frequency Term

Mode = 2

As, it has highest frequency (3)

Solved Questions

Question 1: Study the bar graph given below and find the mean, median, and mode of the given data set.

Solution:

Mean = (sum of all data values) / (number of values)

Mean = (5 + 7 + 9 + 6) / 4  
          =  27 / 2 
          = 6.75

Median = Order the given data in ascending order as: 5, 6, 7, 9

Here, n = 4 (which is even)

Median  =  [(n/2)^th term + {(n/2) + 1}^th term] / 2

Median  = (6 + 7) / 2  
              =  6.5

Mode = Most frequent value 
          = 9  (highest value)

Range = Highest value - Lowest value 

Range = 9 - 5 
           = 4

Question 2: Find the mean, median, mode, and range for the given data

190, 153, 168, 179, 194, 153, 165, 187, 190, 170, 165, 189, 185, 153, 147, 161, 127, 180

Solution:

For Mean:

190, 153, 168, 179, 194, 153, 165, 187, 190, 170, 165, 189, 185, 153, 147, 161, 127, 180

Number of observations = 18

Mean = (Sum of observations) / (Number of observations)

= (190+153+168+179+194+153+165+187+190+170+165+189+185+153+147 +161+127+180) / 18
= 2871/18
= 159.5

Therefore, the mean is 159.5

For Median:

The ascending order of given observations is,
127, 147, 153, 153, 153, 161, 165, 165, 168, 170, 179, 180, 185, 187, 189, 190, 190, 194
Here, n = 18

Median = 1/2 [(n/2) + (n/2 + 1)]^th observation
= 1/2 [9 + 10]^th observation
= 1/2 (168 + 170)
= 338/2
= 169

Thus, the median is 169

For Mode:
The number with the highest frequency = 153
Thus, mode = 153

For Range:
Range = Highest value – Lowest value
= 194 – 127
= 67

Question 3: Find the Median of the data 25, 12, 5, 24, 15, 22, 23, 25

Solution:

25, 12, 5, 24, 15, 22, 23, 25

Step 1: Order the given data in ascending order as: 
5, 12, 15, 22, 23, 24, 25, 25 

Step 2: Check n (number of terms of data set) is even or odd and find the median of the data with respective 'n' value.

Step 3: Here, n = 8 (even) then,

Median = [(n/2)^th term + {(n/2) + 1)^th term] / 2
Median = [(8/2)^th term + {(8/2) + 1}^th term] / 2 
= (22+23) / 2 
= 22.5

Question 4: Find the mode of the given data: 15, 42, 65, 65, 95.

Solution:

Given data set 15, 42, 65, 65, 95
The number with highest frequency = 65
Mode = 65

Unsolved Practice Questions

Question 1: A company recorded the weekly sales (in dollars) of five salespersons as follows: $450, $520, $480, $510, and $490, Find the mean sales value for this group.

Question 2: Find the median of the following data set: 12, 15, 20, 9, 17, 25, 10.

Question 3: A survey collected the number of books read by a group of 10 people last year: 5, 7, 6, 5, 9, 7, 8, 5, 10, 6. What is the mode of the data set?

Question 4: In a classroom, the scores (out of 100) for a test are 56, 78, 67, 45, 56, 90, 56, 67, 78, and 82. Find the mean, median, and mode of the scores.

Question 5: In a skewed distribution, the mean of the data is 40 and the median of the data is 35. Calculate the mode of the data set.

Answer to Practice Questions

• Ans 1: Mean = $490
• Ans 2: Median = 15
• Ans 3: Mode = 5
• Ans 4: Mean = 67.5, Median = 67, Mode = 56
• Ans 5: Mode = 25