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Home > Statistical Methods calculators > Mean, Median and Mode for grouped data example
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Median Example for grouped data
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- Formula & Example
- Mean Example
- Median Example
- Mode Example
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Other related methods
- Mean, Median and Mode
- Quartile, Decile, Percentile, Octile, Quintile
- Population Variance, Standard deviation and coefficient of variation
- Sample Variance, Standard deviation and coefficient of variation
- Population Skewness, Kurtosis
- Sample Skewness, Kurtosis
- Geometric mean, Harmonic mean
- Mean deviation, Quartile deviation, Decile deviation, Percentile deviation
- Five number summary
- Box and Whisker Plots
- Mode using Grouping Method
- Less than type Cumulative frequency table
- More than type Cumulative frequency table
- Class and their frequency table
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3. Median Example
Median of grouped data
Median of discrete frequency distribution
If `n` is odd, then
`M=` value of `((n+1)/2)^(th)` observation
If `n` is even, then
`M=(text{Value of } (n/2)^(th) text{ observation} + text{Value of } (n/2 + 1)^(th) text{ observation})/2`
1. Calculate Median from the following grouped data
Solution:
`x`
`(1)` | Frequency `(f)`
`(2)` | `cf`
`(5)` | | 0 | 1 | 1 | | 1 | 5 | 6 | | 2 | 10 | 16 | | 3 | 6 | 22 | | 4 | 3 | 25 | | --- | --- | --- | | `n=25` | -- |
Median :
M = value of `((n+1)/2)^(th)` observation
= value of `(26/2)^(th)` observation
= value of `13^(th)` observation
From the column of cumulative frequency `cf`, we find that the `13^(th)` observation is `2`.
Hence, the median of the data is `2`.
2. Calculate Median from the following grouped data
| X | Frequency | | 10 | 3 | | 11 | 12 | | 12 | 18 | | 13 | 12 | | 14 | 3 |
Solution:
`x`
`(1)` | Frequency `(f)`
`(2)` | `cf`
`(5)` | | 10 | 3 | 3 | | 11 | 12 | 15 | | 12 | 18 | 33 | | 13 | 12 | 45 | | 14 | 3 | 48 | | --- | --- | --- | | `n=48` | -- |
Median :
M = value of `((n+1)/2)^(th)` observation
= value of `(49/2)^(th)` observation
= value of `24.5^(th)` observation
From the column of cumulative frequency `cf`, we find that the `24.5^(th)` observation is `12`.
Hence, the median of the data is `12`.
Median of continuous frequency distribution
To find Median class, we find cumulative frequencies of all classes and then find `n/2`.
The class whose cumulative frequency is `>= n/2` is called Median class
Median `M = L + (n/2 - cf)/f * c`
where
`:. L = `lower boundary point of median class
`:. n = `Total frequency
`:. cf = `Cumulative frequency of the class preceding the median class
`:. f = `Frequency of the median class
`:. c = `class length of median class
3. Calculate Median from the following grouped data
| Class | Frequency | | 2 - 4 | 3 | | 4 - 6 | 4 | | 6 - 8 | 2 | | 8 - 10 | 1 |
Solution:
Class
`(1)` | Frequency `(f)`
`(2)` | `cf`
`(6)` | | 2-4 | 3 | 3 | | 4-6 | 4 | 7 | | 6-8 | 2 | 9 | | 8-10 | 1 | 10 | | --- | --- | --- | | -- | `n = 10` | -- |
To find Median Class
= value of `(n/2)^(th)` observation
= value of `(10/2)^(th)` observation
= value of `5^(th)` observation
From the column of cumulative frequency `cf`, we find that the `5^(th)` observation lies in the class `4 - 6`.
`:.` The median class is `4 - 6`.
Now,
`:. L = `lower boundary point of median class `=4`
`:. n = `Total frequency `=10`
`:. cf = `Cumulative frequency of the class preceding the median class `=3`
`:. f = `Frequency of the median class `=4`
`:. c = `class length of median class `=2`
Median `M = L + (n/2 - cf)/f * c`
`=4 + (5 - 3)/4 * 2`
`=4 + (2)/4 * 2`
`=4 + 1`
`=5`
4. Calculate Median from the following grouped data
| Class | Frequency | | 0 - 2 | 5 | | 2 - 4 | 16 | | 4 - 6 | 13 | | 6 - 8 | 7 | | 8 - 10 | 5 | | 10 - 12 | 4 |
Solution:
Class
`(1)` | Frequency `(f)`
`(2)` | `cf`
`(6)` | | 0-2 | 5 | 5 | | 2-4 | 16 | 21 | | 4-6 | 13 | 34 | | 6-8 | 7 | 41 | | 8-10 | 5 | 46 | | 10-12 | 4 | 50 | | --- | --- | --- | | -- | `n = 50` | -- |
To find Median Class
= value of `(n/2)^(th)` observation
= value of `(50/2)^(th)` observation
= value of `25^(th)` observation
From the column of cumulative frequency `cf`, we find that the `25^(th)` observation lies in the class `4 - 6`.
`:.` The median class is `4 - 6`.
Now,
`:. L = `lower boundary point of median class `=4`
`:. n = `Total frequency `=50`
`:. cf = `Cumulative frequency of the class preceding the median class `=21`
`:. f = `Frequency of the median class `=13`
`:. c = `class length of median class `=2`
Median `M = L + (n/2 - cf)/f * c`
`=4 + (25 - 21)/13 * 2`
`=4 + (4)/13 * 2`
`=4 + 0.6154`
`=4.6154`
5. Calculate Median from the following grouped data
| Class | Frequency | | 10 - 20 | 15 | | 20 - 30 | 25 | | 30 - 40 | 20 | | 40 - 50 | 12 | | 50 - 60 | 8 | | 60 - 70 | 5 | | 70 - 80 | 3 |
Solution:
Class
`(1)` | Frequency `(f)`
`(2)` | `cf`
`(7)` | | 10 - 20 | 15 | 15 | | 20 - 30 | 25 | 40 | | 30 - 40 | 20 | 60 | | 40 - 50 | 12 | 72 | | 50 - 60 | 8 | 80 | | 60 - 70 | 5 | 85 | | 70 - 80 | 3 | 88 | | --- | --- | --- | | `n = 88` | ----- |
To find Median Class
= value of `(n/2)^(th)` observation
= value of `(88/2)^(th)` observation
= value of `44^(th)` observation
From the column of cumulative frequency `cf`, we find that the `44^(th)` observation lies in the class `30 - 40`.
`:.` The median class is `30 - 40`.
Now,
`:. L = `lower boundary point of median class `=30`
`:. n = `Total frequency `=88`
`:. cf = `Cumulative frequency of the class preceding the median class `=40`
`:. f = `Frequency of the median class `=20`
`:. c = `class length of median class `=10`
Median `M = L + (n/2 - cf)/f * c`
`=30 + (44 - 40)/20 * 10`
`=30 + (4)/20 * 10`
`=30 + 2`
`=32`
6. Calculate Median from the following grouped data
| Class | Frequency | | 20 - 25 | 110 | | 25 - 30 | 170 | | 30 - 35 | 80 | | 35 - 40 | 45 | | 40 - 45 | 40 | | 45 - 50 | 35 |
Solution:
Class
`(1)` | Frequency `(f)`
`(2)` | `cf`
`(7)` | | 20 - 25 | 110 | 110 | | 25 - 30 | 170 | 280 | | 30 - 35 | 80 | 360 | | 35 - 40 | 45 | 405 | | 40 - 45 | 40 | 445 | | 45 - 50 | 35 | 480 | | --- | --- | --- | | `n = 480` | ----- |
To find Median Class
= value of `(n/2)^(th)` observation
= value of `(480/2)^(th)` observation
= value of `240^(th)` observation
From the column of cumulative frequency `cf`, we find that the `240^(th)` observation lies in the class `25 - 30`.
`:.` The median class is `25 - 30`.
Now,
`:. L = `lower boundary point of median class `=25`
`:. n = `Total frequency `=480`
`:. cf = `Cumulative frequency of the class preceding the median class `=110`
`:. f = `Frequency of the median class `=170`
`:. c = `class length of median class `=5`
Median `M = L + (n/2 - cf)/f * c`
`=25 + (240 - 110)/170 * 5`
`=25 + (130)/170 * 5`
`=25 + 3.8235`
`=28.8235`
This material is intended as a summary. Use your textbook for detail explanation.
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