# Grouped Data Standard Deviation Calculator

Created by Michael Darcy
Reviewed by Dominik Czernia, PhD and Alexa Ruiz
Last updated: Oct 09, 2022

If you're not fond of the idea of having to create a frequency distribution table of a dataset from scratch and then having to do deal with all formulae manually, our grouped data standard deviation calculator is here to save the day! It's a tool created specifically to help you compute the mean, variance, and standard deviation of grouped data.

• What is grouped data in statistics;
• How to use the frequency distribution table;
• How to find the variance of grouped data; and
• An example calculation to put all freshly learned knowledge into practice!

However, if you're more interested in ungrouped data, why not try our other tools: population variance calculator and sample size calculator?

## What is grouped data in statistics?

Very often, when we encounter statistics for the first time, we are given some sets consisting of individual data points to work with. However, sometimes it's better to organize those elements, especially if there are many of them. This is what grouped data is in statistics.

In order to sort it, you can create a frequency distribution table. It consists of chosen ranges of grouped data and the number of individual elements lying within this interval - their frequency. Doing this will also make using the grouped data variance calculator easier as its input was based on this type of table.

## How to find the standard deviation of grouped data and mean of grouped data?

Organized data on its own can be informative, but we can usually learn more from it using a little bit of mathematics to analyze it. For example, variance and standard deviation measure the dispersion of the data - how much variation there is from the mean. Although the calculations involved aren't difficult, you may find this part quite laborious, which is why the grouped data standard deviation calculator is here to help you!

What if you wanted to perform the computations yourself? Below you will find how to find the variance of grouped data and, therefore, the standard deviation:

1. Find the midpoint of each range of grouped data. This is done using the formula:
$M = \frac{a+b}{2}$

where:

• $M$ is the midpoint; and
• $a$ and $b$ are the endpoints of the range $a-b$.
1. Determine the number of samples by summing up all frequencies.

2. Calculate the mean of the grouped data, which is the average of the data points. The equation you need is:

$\mu = \frac{\sum(M_i F_i)}{n}$

where:

• $\mu$ represents the mean;
• $M_i$ is the midpoint of the ith range of grouped data;
• $F_i$ is the frequency of the ith range; and
• $n$ is the number of samples found in the previous step.
1. We are ready to find the variance. The standard deviation formula for grouped data is:
$\sigma^2 = \frac{\sum(F_i M_i^2) - (n \mu^2)}{n-1}$

where $\sigma^2$ is the variance.

1. To obtain the standard deviation, take the square root of the variance. Mathematically, we can write this as:
$\sigma = \sqrt{\sigma^2}.$

These quantities can be used for further analysis, for instance, as variables in the relative standard deviation calculator. Knowing this can help establish the quality of data, how much it's changed or what values you could expect in future observations. We could also use the relative frequency calculator for the grouped data to know the probability of their occurrence.

The above steps look quite laborious, do they? Using our grouped data standard deviation calculator is a much faster way to obtain the results!

## Grouped data standard deviation calculator in your daily life - an example

Statistics, especially accompanied by calculations, may not seem overly applicable in everyday life, but that's not necessarily true. This example will show you how to find the standard deviation of grouped data for a case familiar to most of us! For that reason, we will use the grouped data standard deviation calculator.

Suppose that you decide to lose some weight but you're also an avid coffee drinker who likes various types of caffeinated drinks and doesn't wish to give them up. That's not an issue; you can always turn your steps into burned calories - but how much should you walk? Well, after a month of noting your orders, you have 30 observations. This should be enough to give you a reasonable estimate but too much to handle separately. You could make a small :

Kcal

Frequency

100-129

5

130-159

4

160-189

12

190-219

6

220-249

3

Inputting these into the grouped data variance calculator, you find that, on average, your coffee has 172.5 kcal (mean), but the actual value is likely to vary by 36 kcal (standard deviation).

## FAQ

### What is the difference between standard deviation and variance?

Variance uses squares, whereas the standard deviation is expressed in the same units as the mean, so the latter is easier to interpret and use. However, you must calculate the variance to obtain the standard deviation.

### Why does the variance use squares?

Variance uses squared differences from the mean to prevent obtaining negative values or zero due to canceling out the terms. It also weighs outliers more heavily than data closer to the average, so the differences are more significant.

### How to find the midpoint of an interval?

The midpoint of an interval is an average of the interval's lower and upper limits. To find it:

1. Add the lower limit to the upper limit.
2. Divide this sum by 2. The result is the midpoint.

### How to calculate the mean of grouped data?

1. Determine the midpoint of each interval.
2. Sum up all frequencies to obtain the number of samples.
3. For each range of grouped data, multiply its midpoint by associated frequency. Add them all up.
4. Divide this value by the number of samples.
Michael Darcy
You can enter up to 25 values of ranges and corresponding frequencies (new rows will appear).
The range should be in the X-Y format, e.g., 3-11, 0.3-0.7, (-4)-(-1), -5--8.
Range₁
Frequency₁
Range₂
Frequency₂
Results
 Mean (μ): 0 Standard deviation (σ): 0 Variance (σ²): 0
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