The arithmetic mean of a given data is the sum of all observations divided by the number of observations. For example, a cricketer's scores in five ODI matches are as follows: 12, 34, 45, 50, 24. To find his average score in a match, we calculate the arithmetic mean of data using the mean formula:
Raw Data
Example: If the heights of 5 people are 142 cm, 150 cm, 149 cm, 156 cm, and 153 cm. Find the mean height.
Mean height, x̄ = (142 + 150 + 149 + 156 + 153)/5
= 750/5
= 150
Mean, x̄ = 150 cm
Thus, the mean height is 150 cm.
When the data is present in tabular form, we use the following formula:
Mean, x̄ = (x_{1}f_{1} + x_{2}f_{2} + ... + xnfn)/(f_{1} + f_{2} + ... + f_{n})
Consider the following example.
Example 1: Find the mean of the following distribution:
Calculation table for arithmetic mean:
Mean, x̄ = (∑x_{i} f_{i})/(∑fi)
= 360/40
= 9
Thus, Mean = 9
Example 2: Here is an example where the data is in the form of class intervals. The following table indicates the data on the number of patients visiting a hospital in a month. Find the average number of patients visiting the hospital in a day.
In this case, we find the classmark (also called as midpoint of a class) for each class.
Note: Class mark = (lower limit + upper limit)/2
Let x_{1}, x_{2}, x_{3} , . . . , xn be the class marks of the respective classes.
Hence, we get the following table:Mean, x̄ = (∑x_{i} f_{i})/(∑fi)
= 860/30
= 28.67
x̄ = 28.67
Challenging Question:
Let the mean of x_{1}, x_{2}, x_{3} … x_{n} be A, then what is the mean of:
(x_{1} + k), (x_{2} + k), (x_{3} + k), … , (x_{n }+ k)
(x_{1}  k), (x_{2 } k), (x_{3}  k), … , (x_{n}  k)
kx_{1}, kx_{2}, kx_{3}, … , kx_{n}
The value of the middlemost observation, obtained after arranging the data in ascending or descending order, is called the median of the data.
For example, consider the data: 4, 4, 6, 3, 2. Let's arrange this data in ascending order: 2, 3, 4, 4, 6. There are 5 observations. Thus, median = middle value i.e. 4.
Case 1: Ungrouped Data
To find the median, we need to consider if n is even or odd. If n is odd, then use the formula:
Median = (n + 1)/2th observation
Example 1: Let's consider the data: 56, 67, 54, 34, 78, 43, 23. What is the median?
Arranging in ascending order, we get: 23, 34, 43, 54, 56, 67, 78. Here, n (number of observations) = 7
So, (7 + 1)/2 = 4
∴ Median = 4th observation
Median = 54
If n is even, then use the formula:
Median = [(n/2)th obs.+ ((n/2) + 1)th obs.]/2
Example 2: Let's consider the data: 50, 67, 24, 34, 78, 43. What is the median?
Arranging in ascending order, we get: 24, 34, 43, 50, 67, 78.
Here, n (no.of observations) = 6
6/2 = 3
Using the median formula,
Median = (3rd obs. + 4th obs.)/2
= (43 + 50)/2
Median = 46.5
Case 2: Grouped Data
When the data is continuous and in the form of a frequency distribution, the median is found as shown below:
where,
Let's consider the following example to understand this better.
Example: Find the median marks for the following distribution:
We need to calculate the cumulative frequencies to find the median.
Calculation table:
N/2 = 50/2 = 25
Median Class = (20  30)
l = 20, f = 22, c = 14, h = 10
Using Median formula:
= 20 + (25  14)/22 × 10
= 20 + (11/22) × 10
= 20 + 5 = 25
∴ Median = 25
Case 1: Ungrouped Data
For example in the data: 6, 8, 9, 3, 4, 6, 7, 6, 3, the value 6 appears the most number of times. Thus, mode = 6. An easy way to remember mode is: Most Often Data Entered. Note: A data may have no mode, 1 mode, or more than 1 mode. Depending upon the number of modes the data has, it can be called unimodal, bimodal, trimodal, or multimodal.
The example discussed above has only 1 mode, so it is unimodal.
Case 2: Grouped Data
When the data is continuous, the mode can be found using the following steps:
where,
Consider the following example to understand the formula.
Example: Find the mode of the given data:
The highest frequency = 12, so the modal class is 4060.
 l = lower limit of modal class = 40
 fm = frequency of modal class = 12
 f_{1} = frequency of class preceding modal class = 10
 f_{2} = frequency of class succeeding modal class = 6
 h = class width = 20
Using the mode formula,
= 40 + (2/8) × 20
= 45
∴ Mode = 45
Median formula for grouped data: Median =
where,
Mode formula for ungrouped data: Mode = Observation with maximum frequency
Mode formula for grouped data: Mode =
where,
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