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Arithmetic mean of the interval distributive series

Calculator of the arithmetic mean of a series of distributing class intervals.

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

Median of the interval distributive series

Arithmetic mean of the interval distributive series - calculator

Arithmetic mean - information

Arithmetic mean is the value of a specific feature $\bar x $ that would be assumed by all observations (statistical population units) if there were no differences between them due to the level of the studied feature.

We do not use the arithmetic mean when:

Condition: The arithmetic mean for a distributive series with open class intervals can be calculated if the class intervals have small numbers (practically 5% of the total number). If this condition is met, the intervals can be closed, i.e. the upper limits of the class intervals can be assumed. If this condition is not met, the ranges must not be closed and the mean must not be calculated.

2. The unusual values appear in a series

3. If the community is heterogeneous

The formula used to calculate the arithmetic mean of the interval series:

$$\bar x = \frac{\dot x_1 n_1+ \dot x_2 n_2 + ... + \dot x_i n_i}{n_1+n_2+... + n_i}=\frac{\sum{\dot x_i\,n_i}}{\sum{n_i}}$$ where
$\dot x_i = \frac{x_{0i}+x_{1i}}{2}$ -middle of the i-th class intervals
$n_i $ -frequency (size) of the i-th class intervals

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	35
15-20	55
20-35	30
25-30	20
30-35	15

The task is to calculate the arithmetic mean of the given distribution series.

First, let's define the statistical population, statistical unit and variable.
Employees constitute the statistical population. The statistical unit is one employee. A feature of a continuous quantitative variable is the distance to work.

From the first row of the data table, we read that 5 employees travel from 0 to 5 kilometers to work. The second line says 25 workers live between 5 and 10 km, etc ..

From the formula for the arithmetic mean of the interval series, we know that we need to determine the midpoints of the intervals.
So the middle of the first interval will be equal to: $\dot x_1 = \frac{x_{01}+x_{11}}{2} = \frac{0+5}{2}= 2,5$. In the same way, we calculate the centers of successive intervals.

Then, for each line we calculate the product of frequency and the middle of the range, e.g. the first line- $\dot x_1 · n_1 = 2,5 · 5 = 12,5$. We repeat the calculations for the remaining lines.

Distance of residence in km. $ (x_{0i}-x_{1i})$	Number of employees $(n_i) $	$\dot x_i $	$\dot x_i · n_i $
0-5	5	2,5	12,5
5-10	25	7,5	187,5
10-15	35	12,5	437,5
15-20	55	17,5	962,5
20-35	30	22,5	675
25-30	20	27,5	550
30-35	15	32,5	487,5
∑	185	-	3312,5

Substituting the sums into the formula we can calculate the arithmetic mean:

$$\bar x = \frac{\sum{\dot x_i\,n_i}}{\sum{n_i}}=\frac{3312,5}{185}=17,9054$$

Interpretation: The average distance between workers' residence and work is 17.91 km.

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