Mean
2026-02-28 17:21 Diff
https://brightchamps.com/en-us/math/data/mean

Mean is the average value of a set of numbers. To find the mean, we add all the values and divide the sum by the total number of values. Depending on the data type, we use different methods to find the mean. 

Mean for Ungrouped Data

Ungrouped data refers to raw data that has not been organized into groups, classes, or intervals. To find the mean, add all the values and divide the sum by the number of values. 
 

\(\bar{x} = \frac{\sum x}{n} \)


For example, find the mean of 4, 6, 8, 10. 
 

\({\bar x} = {{4 + 6 + 8 + 10} \over 4}\)
 

\(= {28 \over 4} = 7\)

Mean for Grouped Data
Grouped data refers to data organized into groups or class intervals. The mean for grouped data can be calculated using three methods: 
 

  • Calculating Mean Using the Direct Method
  • Calculating Mean Using Assumed Mean Method
  • Calculating Mean Using Step Deviation Method

Calculating Mean Using the Direct Method
In the direct method, the mean for grouped data is calculated by multiplying each class midpoint by its frequency, summing the products, and dividing by the total frequency.
 

\(\bar{x} = \frac{\sum f x}{\sum f} \)

For example, find the mean of the given data set. 

Class Interval Frequency Midpoint (x) fx 0 - 10 4 5 20 10 - 20 6 15 90 20 - 30 5 25 125

\(Σfx = (4 × 5) + (6 × 15) + (5 × 25) = 235 \)
 

\(Σf = 4 + 6 + 5 = 15 \)
 

\(\bar{x} = \frac{\sum f x}{\sum f} \)

\(\bar x = {235 \over 15}\)

= 15.67.

Calculating Mean Using Assumed Mean Method

The assumed mean method is used when the sample size is large. We take a mean (A) and calculate deviations from it. 


Here, \( \bar{x} = A + \frac{\sum f d}{\sum f} \)

where, d = x - A

For example,

x f d = x - A 10 2 -10 20 3 0 30 5 10

 Let’s assume A = 20

Then, 

\(Σfd = (-10 × 2) + (0 × 3) + (10 × 5) \)
 

\(= -20 + 0 + 50 \)
 

= 30
 

\({\bar x} = 20 + {30 \over 10}\)
 

\(= 20 + 3 = 23\)

Calculating Mean Using Step Deviation Method
The step deviation method is used when class intervals are equal and values are significant. The formula to find the mean using the step deviation method is: 
 

\(\bar{x} = A + h \frac{\sum f _u}{\sum f} \)
 

where A = assumed mean

h = class width

\(u = \frac{x - A}{h} \)

For example, 

Class Interval f Midpoint(x) u \(f_u\) 10-20 5 15 -1 -5 20-30 7 25 0 0 30-40 8 35 1 8

Here, A = 25

h = 10

\(Σf_u = 3 \)

\(Σf = 20 \)

Substitute the values in the equation: 

\(\bar{x} = 25 + 10 ( \frac{30}{20}) \)

\({\bar x}= 25 + 1.5 = 26.5 \)

Mean of Negative Numbers 

The method for finding the mean remains the same even when the numbers are negative. This means the mean of negative values is simply the sum of all the observations divided by the total number of observations.

For example, -5, 10, -3
 

\({\bar x} = {{-5 + 10 - 3} \over 3}\)
 

\(= {2 \over 3}\)
 

= 0.667.