Median for grouped data - Guide step by step and examples

This article will be about How to find the median for grouped data, with the formulas and examples solved step by step.

The median in a data set is the number that is in the middle of all the elements when the set is ordered, either, from highest to lowest, or from lowest to highest, the median will be the same in both cases. When we have individual data, it is pretty simple to define the median, because we only have to order the elements of the set and then look for the middle number, but, when we want to find the median in grouped data, we cannot order individual numbers because we have intervals, so the process to find the median will change.

In simple terms, the median is the central number of a data series, the inconvenient with the median for grouped data is that we do not have any number, for example, we could know that the interval ]10,20] has a frequency of 4, this means that there are 4 numbers between 10 and 20, but we do not know which numbers are these, or if they repeat, so, what we do to find the median for grouped data is determine an estimated data where the median could be, so it is possible that the calculated median is not even a number between the compiled data, but even so we are going to take this value as the median.

Formula of the median for grouped data

Median = L_i *

n/2
- F_i-1 /f_i
* A

As we can see in the median for grouped data formula, there are many variables that we have to find to use this formula and calculate the median, this is why we are going to define each of this variables and how to find them, but, if it is complicated to understand the theory, we recommend to go to the examples section where we practice all this.

n/2: In this part of the formula, the letter “n” is the sum of all the frequencies, and this result will show where the median is, the median will be in the next interval where the absolute frequency is greater than “n/2”.

F_i-1: the capital “F_i” is the result of the cumulative frequency of the interval of the median. The cumulative frequency is obtained by adding all the previous frequencies, while the i-1 means that is the previous cumulative frequency to the median interval.

Frequency	Cumulative frequency
2	2
1	3
4	7
5	12

L_i: This is the lower limit of the interval where the median is located. The lower limit of an interval is the lower number of the interval, for example in ]10,20], the lower limit of this interval is the number 10.

f_i: the lowercase “f” is the frequency of the interval, and “i” is the interval of the median, so this is the frequency of the interval where the median is located.

A: The capital “A” is the width of the interval, this is obtained by subtracting the upper limit minus the lower limit of the interval of the median, for example, the width of the interval ]30,50] is: 50-30=20.

Examples of the median for grouped data

Example 1: A poll was made where they were asking about the hours that people spend on their phones, calculate the median of the results.

Hours	Frequency
]0-2]	28
]2-4]	160
]4-6]	42
]6-8]	2
Total	232

First we are going to find every part of the formula separately and then we are going to put everything together.

First we find the median interval
n/2
232/2
R// 116

With this results we can see in which interval the median is, as we said, the median is in the interval that has a greater frequency than n/2, so now we proceed to find the accumulative frequency.

(With * the median interval)

Hours	Frequency (f_i)	Accumulative frequency( F_i)
]0-2]	28	28
]2-4]	160	*188
]4-6]	42	230
]6-8]	2	232

Now, we can also find F_i-1 that would be the accumulative frequency of the first interval, because is the previous interval of the median interval, that would be 28, also, we know that the L_i of the interval of the median is 2.

And last we find “A”.

Now we find the with of the interval ]2 - 4]
A = 4 - 2
A = 2

And now we find the median
Median = L_i +

n/2
- F_i-1 /f_i
* A
Median = 2 +

232/2
- 28 /160
* 2
Median = 2 +
116 - 28 /160
* 2
Median = 2 +
88 /160
* 2
Median = 2 + 0.55 * 2
Median = 2 + 1.1
Median = 3.1

Example 2: Find the median of the following data

Intervals	Frequency
]10 - 20]	91
]20 - 30]	88
]30 - 40]	35
]40 - 50]	102
Total	316

First we find the median
n/2
316/2
R// 158

Now we find the accumulative frequency and we find the interval that contains the median

(With * the median interval)

Intervals	Frequency	Accumulative frequency
]10 - 20]	91	91
]20 - 30]	88	*179
]30 - 40]	35	214
]40 - 50]	102	316
Total	316

Now we find the interval width

interval of the median ]20 - 30]
A = 30-20
A = 10

With this we know that the L_i is 20, that F_i is 91 and that A is 10, so we are going to put all this in the formula.

And finnaly we find the median
Median = L_i +

n/2
- F_i-1 /f_i
* A
Median = 20 +

316/2
- 91 /88
* 10
Median = 20 +
158 - 91 /88
* 10
Median = 20 +
67 /88
* 10
Median = 20 + 0.76 * 10
Median = 20 + 7.6
Median = 27.6

Example 3: Somebody asked to a group of people the percentage of monthly savings, find the median with the results.

Percentages	Frequency
]0 - 5]	31
]5 - 10]	201
]10 - 15]	12
Total	244

First we find every part of the formula and then we solve it

Find the interval of the median
n/2
244/2
R// 122

Now we find the accumulative frequency of the interval median

(With * the interval of the median)

Percentages	Frequency	Accumulative frequency
]0 - 5]	31	31
]5 - 10]	201	*232
]10 - 15]	12	244
Total	244

Then we find the interval width

Interval: ]5 - 10]
A = 10-5
A = 5

With this we now that L_i = 5, f_i = 201, F_i-1 = 31 y A = 5, And now we can calculate the median

Calculate the median
Median = L_i +

n/2
- F_i-1 /f_i
* A
Median = 5 +

244/2
- 31 /201
* 5
Median = 5 +
122 - 31 /201
* 5
Median = 5 +
91/201
* 5
Median = 5 + 0.45 * 5
Median = 5 + 2.25
Median = 7.25

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