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
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.
↪ It is denoted by the symbol , read as .
↪ For n observations
Mean,
↪ The Greek symbol (for the letter Sigma) is used for summation.
Instead of writing , we write , which is read as ‘the sum of x_{i} 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 , ,
Then, total of
Total of
…
Total of
Sum of all observations
Total number of Observations
So, the Mean,
Example : Find the mean salary of 60 workers of a factory from the following table:
Solution :
Mean Salary = ₹= ₹ 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 = th term
↪ For even number of data,
Median = Mean of the th term & th term
↪ Median in above example –
Number of terms = 60 (even)
th term = 30th term = 5000
th term = 31th term = 5000
∴ Median salary =
= ₹ 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 ⏩ |