# Dispersion Calculator

Created by Komal Rafay
Reviewed by Dominik Czernia, PhD and Jack Bowater
Last updated: Oct 31, 2022

The dispersion calculator is here to calculate the spread of your data to tell you how scattered or squeezed your data actually is.

Today is your lucky day! This calculator has everything you need to learn what the measure of dispersion is, how the measures of dispersion helps you interpret data, and how to calculate the dispersion. If you are even more curious and want to know what dispersion means in statistics and why dispersion is calculated, we have it all explained below.

## Dispersion calculator - what dispersion means in statistics

Our measures of dispersion calculator is a handy tool that allows you to calculate the dispersion of your collected data to help you understand how squeezed or spread out your data is. In other words, it tells you how far the numbers are spread out from the average value.

The only thing you need to calculate the dispersion using this tool is the data you have collected. Just remember to specify whether the data is for an entire population or a sample.

Once you input your data set, you get the following list of results:

• Population size;
• Mean;
• Median;
• Mode;
• Minimum value;
• Maximum value;
• Range;
• Midrange;
• Variance;
• Standard deviation;
• First quartile;
• Third quartile;
• Interquartile range; and
• Quartile deviation.

These values allow the spread and dispersion of the data to be interpreted. In this case, the dispersion definition is the measure of the spread of data.

Statistical dispersion means the degree to which a set of numerical data is likely to deviate from an average value or its mean.

## Why is dispersion calculated?

By now, you have understood how to calculate the dispersion of data, but some of you might still be wondering why is dispersion calculated?
The significance of knowing what dispersion in statistics is is as follows:

1. Calculating dispersion helps us understand the spread of the data.

2. It is one of the most important measures of frequency distribution.

3. Dispersion is the basis of comparison between two or more frequency distributions. This is due to the fact that multiple distributions can have the same average but have significant differences in their variability.

## How to calculate dispersion in statistics?

In statistics, the dispersion of a data set can be calculated based on various values extracted from the data set. These values are called the measures of dispersion and they help you understand the spread of your data, i.e., how squeezed or scattered the data actually is.

• Dispersion from range

The range of data is the difference between its minimum and maximum value.

The value of the midrange can be obtained by dividing the range by two or, in other words, calculating the average of the maximum and minimum values.

To calculate dispersion from range, the formula is the following:

$\text {range = max - min}$

where:

• $\text {max}$ - The maximum value of the data set; and

• $\text {min}$ - The minimum value of the data set.

• Dispersion from interquartile range and quartile deviation

The quartiles divide the numerical data into four equal parts using three values: Q1, Q2, and Q3.

You have to arrange your data in ascending order to calculate the quartiles. Q2 is the median of the data, Q1 is the median of the lower half of the data, and Q3 is the median of the upper half of the data set.

The interquartile range (IQR) is also known as the midspread. We calculate it as the difference between the third and first quartile, which are also the 75th and 25th percentile, respectively.

The formula to calculate dispersion from the interquartile range is:

$\text {IQR} = \text Q_3 - \text Q_1$

where:

• $\text {IQR}$ - Interquartile range;

• $\text Q_3$ - Third quartile; and

• $\text Q_1$ - First quartile.

The quartile deviation is calculated by dividing the interquartile range by two.

$\text {Quartile deviation = IQR / 2}$

• Dispersion from variance

Variance is the measure of the variability of data.

To calculate variance, first deduct the mean from each value in the data set. Second, square the deducted values. Third, find the sum of the squared values and, finally, divide the sum by the number of values or population size.
You may want to check out our variance calculator to get to know more about variance in particular.

Its value can also be obtained by squaring the standard deviation.

The formula to calculate dispersion from variance is:

$\sigma^2 = \sum (\text x - \mu)^2 / \text N$
where:

• $\sigma^2$ - Variance;

• $\text x$ - Each value in the data set;

• $\mu$ - Population mean; and

• $\text N$ - Population size or the number of values.

• Dispersion from standard deviation

This is one of the most common measures of dispersion.

Standard deviation is the square root of the sum of all squared deviations from the mean divided by the number of observations. And the standard deviation calculator can tell you all about it.

It is used more frequently because it uses all the values of the data and so is a good representation of the entire dataset.
Standard deviation is also summarized as the square root of the variance, and its formula is:

$\text {SD} = \sqrt \sigma$

where:

• $\text {SD}$- Standard deviation; and

• $\sigma$- Variance.

• Dispersion from mean

Mean is the average of the data set and is calculated as:

$\text A= \sum \text x / \text n$

where:

• $\text A$ - Arithmetic mean;

• $\sum \text x$- Sum of the data set; and

• $\text n$ - Number of values or population size.

In our measures of dispersion calculator, we calculate all of the above metrics at once. But in case you want to take a look at them individually, we suggest you take a look at our other statistics calculators, like:

## Importance of measures of dispersion in statistics

The importance of measures of dispersion in statistics can be realized in scenarios where we need to draw a conclusion from our initial data as they allow you to identify the margin of error you are allowed to work with. They also show you the variability of your data.

The greater your dispersion, the less representative your central tendency is. Check the below FAQ section to see some dispersion examples.

## FAQ

### What is the first quartile of 9, 78, 23, 4, 5, 76, 3, 10?

4.5 is the first quartile of 9, 78, 23, 4, 5, 76, 3, 10.

The first quartile is the median of the first half of the data.

Remember, to calculate quartiles, you need to arrange your data in ascending order.

### How to calculate dispersion from standard deviation?

Standard deviation is the square root of the variance, and its formula is:

SD = √σ

where:

• SD - Standard deviation; and

• σ - Variance.

It is one of the most common measures of dispersion and is used more frequently because it is calculated using all the data values.

### How can I calculate dispersion in statistics?

To calculate the dispersion, you may use any of the measures of dispersion: variance, mean or mean deviation, quartile or quartile deviation, and standard deviation.

Dispersion is calculated when you have to study the spread of your data in terms of how spread out or squeezed it is from its average value.

Dispersion also signifies the frequency distribution of the data.

### What is the standard deviation of the values 23, 45, 67, 87, 98, 34, 11, 76?

The standard deviation of the values is 29.49.

Standard deviation is the square root of the variance, and its formula is:

SD = √σ

where:

• SD - Standard deviation; and

• σ - Variance

### What is the standard deviation of a population if the variance is 182.2?

The standard deviation is 13.5 if the variance is 182.2.
Standard deviation is the square root of the variance, which makes variance the square of standard deviation.

To calculate variance, follow the steps below:

1. Deduct the mean from each value in the data set.
2. Square the deducted values.
3. Find the sum of the squared values.
4. Divide the sum by the number of values or population size.
Komal Rafay
You may enter up to 50 values:
x₁
x₂
Settings:
Data type
Sample
Results:
 Population size: 0 Mean: Median: Mode(s): Range: Midrange: Minimum value: Maximum value: Variance: Standard Deviation: First quartile: Third quartile: Interquartile range: Quartile deviation:
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