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Median of the distributive series of class intervals - calculator

With this statistical calculator, you will learn how to calculate the median of an interval distributive series from grouped data.

To get the result, enter the intervals separated by "-" eg 10-20 and frequency.

See also:

Quartiles Q1, Q2, Q3 of the interval series

Mode of the group series of distributing class intervals

Arithmetic mean of the interval distributive series

Median of the interval distributive series

Median information

Median divides the community into two equal parts.
Half of the units have trait values less than or equal to the median and half have the trait values greater than the median.
Median, next to the arithmetic mean, is the most frequently used statistical parameter. It can be calculated in cases where it is impossible to calculate the mean (e.g. open class intervals) and also the modal - different spans of class intervals.

Formula for the median:

$$Me=x_0 + (N_{Me} - n_{isk-1}) · \frac{h_{Me}}{n_{Me}}$$

N_Me = n/2 - number (position) for the median
n_Me - frequency of the range containing the median
x₀ - lower class boundary of the range containing the median
n_isk-1 - cumulative frequency range before the cumulative frequency of the median
h_Me = x_1Me(higher) - x_0Me(lower) - size (width) of the range containing the median

EXAMPLE

In one of the companies, the distance between the employees' place of residence and the workplace was examined, the data was as follows:

Distance of residence in km. $ (x_{0i}+x_{1i})$	Number of employees $(n_i) $
0-5	5
5-10	25
10-15	30
15-20	55
20-35	30
25-30	20
30-35	15

The task is to calculate the median of the given distribution series.

First, we should calculate the cumulative frequencies. In our example, we add the values from the Number of employees column one by one. The calculations are presented in the table below.

Distance of residence in km. $ (x_{0i}+x_{1i})$	Number of employees $(n_i) $	Cumulative frequency $(n_{isk})$
0-5	5	0+5=5
5-10	25	5+25=30
10-15	30	30+30=60
15-20	55	60+55=115
20-35	30	115+30=145
25-30	20	145+20=165
30-35	15	165+15=180

Now we can proceed to the calculation of the position (number) of the median
$$N_{Me}=\frac{n}{2}$$
where n is the size of surveyed population. In our example, it is the number of all employees, n = 180, i.e. even (that's why we used the formula for even observations). Substituting to the formulas we get:

$$N_{Me}=\frac{180}{2}=90$$

To check to which range the median belongs, in the Cumulative Frequencies column we look for the first value greater or equal to the median position. In our task it will be the value 115. Since 115 is the first cumulative frequency greater than 90.
So we see that the median belongs to the range of 15-20 km, the frequency of which (number of employees) is 55.

We already know to which range the median belongs, but we do not know its exact value. To calculate the median value of the interval series, we will use the formula:

$$Me=x_0 + (N_{Me} - n_{isk-1}) · \frac{h_{Me}}{n_{Me}}$$

where:
x₀ = 15 - lower class boundary of the range containing the median. Earlier, we established that this interval is 15-20 km, i.e. its lower limit will be 15.
N_Me = 90 - position (number) of the median, which we calculated earlier from the formula $ N_{Me}=\frac{n}{2} $
n_isk-1 = 60 - cumulative frequency of the range that precedes the cumulative frequency to which the median belongs. We determined that the cumulative frequency to which the median belongs is 115. Looking at the row above in the table, we can see that the preceding cumulative number is 60.
h_Me = x_1Me(higher) - x_0Me(lower) = 20-15 = 5 - size (width) of the range containing the median. The range of the median is 15-20 km, so to calculate the width from the upper value, we subtract the lower value.
n_Me = 55 - frequency of the range (number of employees) containing the median.

Now just substitute the values into the formula:

$$Me=15 + (90 - 60) · \frac{5}{55} = 17,73 $$

Interpretation: 50% of employees live 17.73 km or less, and the remaining 50% live more than 17.73 km from their workplace.

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