Measure of Central Tendency for Ungrouped Data – Mean/Median/Mode

Here we’ll learn to find measure of central tendency for ungrouped data. Mean, Mode & Median are different central values used to represent set of data.

Prerequisite / Revise this :

Representative or Central value

Measure of central tendency for ungrouped data

It is a single value that is used to describe a set of data by identifying the central position within that set of data.

↪ Ideally, the central value is the score that is the best representative value for all of the individuals in the distribution.

↪ Central value lies between minimum and maximum values of a data.

↪ Different forms of data need different forms of representative or central value to describe it.

Measure of central tendency for ungrouped data – Mean, Mode and Median

Range

  • The difference between the highest and the lowest observation of a data is called Range of the data.
  • It gives us an idea of the spread of the observations.
  • Arrange the data in ascending order, then find the difference between highest and lowest observation.

Mean/Average

Mean of a data is defined as the ratio of the sum of all observations to the number of observations.

Mean=\frac{Sum\, of\, all\, observations}{Number\, of\, observations}

↪ It is denoted by the symbol \bar {x}, read as 'x\, bar'.

↪ For n observations

Mean, \bar{x} = \frac{x_1 + x_2 + \cdots + x_n}{n}

\bar{x} = \frac{\sum_{i=1}^n f_i x_i}{n}

↪ The Greek symbol \sum (for the letter Sigma) is used for summation.

Instead of writing x_1 + x_2 + \cdots + x_n, we write \sum_{i=1}^n x_i, which is read as ‘the sum of xi as i varies from 1 to n’.

Try This

[ Measure of central tendency for ungrouped data Examples ]

Q) A cricketer scores the following runs in eight innings:

58, 76, 40, 35, 46, 45, 0, 100.

Find the range of scores and the mean score.

Solution

Arranging the data in ascending order,

0, 35, 40, 45, 46, 58, 76, 100

Range of runs scored = highest score – lowest score
= 100 – 0 = 100

Total run scored = 58+76+40+35+46+45+0+100 = 400

Number of innings = 8

Mean score of eight innings = 400/8 = 50

Finding Mean from Frequency distribution table

Let, the frequency of x_1 \,be\, f_1 , x_2\, be \,f_2,\cdots , x_n \,be \,f_n

Then, total of x_1 =f_1\times x_1
Total of x_2 =f_2\times x_2

Total of x_n =f_n\times x_n

Sum of all observations

=f_1\times x_1 + f_2\times x_2 +\cdots + f_n\times x_n

=\sum_{i=1}^n f_i x_i

Total number of Observations
=f_1 + f_2 +\cdots + f_n

=\sum_{i=1}^n f_i

So, the Mean,

\bar{x} = \frac{\sum_{i=1}^n f_i x_i}{\sum_{i=1}^n f_i}

Example : Find the mean salary of 60 workers of a factory from the following table:

Measure of central tendency for ungrouped data

Solution :

Measure of central tendency for ungrouped

Mean Salary = ₹\frac{3,05,000}{60}= ₹ 5,083.33

Mode

The mode is that value of the observation which occurs most number of times, i.e., an observation with the maximum frequency is called the mode.

↪ This measure of central tendency is very useful for the apparel and shoe industries .
Using the knowledge of mode, these industries decide which size of the product should be produced in large numbers.

↪ Modal salary in above example is ₹ 3000.

Median

Median refers to the value which lies in the middle of the data (when arranged in an increasing or decreasing order) with half of the observations above it and the other half below it.

↪ It divides the data in two group with equal number of observations.

↪ For odd number of data,

Median = \left[\frac{n+1}{2}\right]th term

↪ For even number of data,

Median = Mean of the \frac{n}{2} th term & \frac{n}{2} + 1 th term

↪ Median in above example –

Number of terms = 60 (even)

\frac{n}{2} th term = 30th term = 5000

\frac{n}{2} + 1 th term = 31th term = 5000

∴ Median salary = \left(\frac{5000+5000}{2}\right)

= ₹ 5000

Measure of central tendency for ungrouped data Examples

Try These

Q1) Find the median of → 2, 6, 5, 3, 0, 3, 4, 3, 2, 4, 5,2, 4,

Q2) Which of the central representative value is appropriate in the following cases –

(i) We have to decide upon the number of chapattis needed for 25 people called for a feast.

(ii) A shopkeeper selling shirts has decided to replenish her stock.

(iii) We need to find the height of the door needed in our house.

Solution

A1) Arranging the data in ascending order,

0, 2, 2, 2, 3, 3, 3, 4, 4, 4, 5, 5, 6

The median of the data = 3

A2)

(i) Mean, product of mean and number of people would give total number of chapattis needed.

(ii) Mode, most sold shirt type should be replenished.

(ii) No central representative value is appropriate in this case, we need largest value to let tallest person pass through the door.

Measure of Central Tendency for Ungrouped Data  – Problems with Solutions

⏪ Graphical Representation of Data Measures of Central Tendency for Grouped Data ⏩

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