# Standard Deviation Calculator

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Standard Deviation Formula

How to calculate sample standard deviation (step-by-step example)

How to calculate population standard deviation (example)

Youtube explanation of standard deviation

What does standard deviation mean (or tell us)?

What are uses for standard deviation?

## Standard Deviation Formula

where,

∑ = The sume of each calculation; (x

_{i}- μ)

^{2}for each value i

x

_{i}= Value for each number within the dataset (i just represents the value)

μ = The average for the values in the dataset

n = The number of values in the dataset, if you have 10 numbers in the dataset, then n = 10

σ = Standard Deviation

## Standard Deviation of a Sample

You have collected the following data from an experiment:

23, 34, 14, 23, 12, 33, 13, 16

### Step 1. Find the Mean

This value is also known as your average or μ

23 + 34 + 14 + 23 + 12 + 33 + 13 + 16 = 168 ÷ 8 = 21### Step 2. Establish Deviations

This is completed by subtracting your average (or μ )from each of your values from the data set

1. (23 - 21) = 22. (34 - 21) = 13

3. (14 - 21) = -7

4. (23 - 21) = 2

5. (12 - 21) = -9

6. (33 - 21) = 12

7. (13 - 21) = -8

8. (16 - 21) = -5

### Step 3. Square your deviations

1. 2² = 42. 13² = 169

3. -7² = 49

4. 2² = 4

5. -9² = 81

6. 12² = 144

7. -8² = 64

8. -5² = 25

### Step 4. Add the values you just squared to find their sum

4 + 169 + 49 + 4 + 81 + 144 + 64 + 25 = 540### Step 5. Divide the sum by the number of values minus 1

There are 8 values in the set, so you will divide by 7(N-1)

540 ÷ 7 = 77.1429### Step 6. Find the square root

√ 77.1429 = 8.7831You have just solved a standard deviation problem!...Now to tackle how to understand what it means.

## Population Standard Deviation

It is important to note that finding the population standard deviation has the same exact formula as the sample standard deviation, with one exception. Rather than subtracting 1 from n (number of values in the denominator), you only divide by n.

The population standard deviation formula is:

_{i}-μ)

^{2}/ n

An example of a problem to find the standard deviation of a population is as follows:

Presume our data consists of the following ten values:

2, 3, 5, 5, 6, 8, 8, 9, 9, 10

### Step 1. Find the Mean

To find this we find the sum of the values and divide by 102, 3, 5, 5, 6, 8, 8, 9, 9, 10 ÷ 10 = 6.5

The mean of these ten values is 6.5

### Step 2. Find the Difference and Square them

To find the standard deviation of the population you must find the difference of each value by subtracting it from the mean and squaring the result of each value.i.e.

(2-6.5)= (-4.5)^2= 20.25 (3-6.5)= (-3.5)^2= 12.25

(5-6.5)= (-1.5)^2= 2.25 (5-6.5)= (-1.5)^2= 2.25

(6-6.5)= (-.5)^2= .25 (8-6.5)= (1.5)^2= 2.25

(8-6.5)= (1.5)^2= 2.25 (9-6.5)= (2.5)^2= 6.25

(9-6.5)= (2.5)^2= 6.25 (10-6.5)= (3.5)^2= 12.25

### Step 3. Find the sum of values from Step 2 and Divide

Next you will need to determine the average of the sum of these values and then determine the square root:(20.25 + 12.25 + 2.25 + 2.25 + .25 + 2.25 + 2.25 + 6.25 + 6.25 + 12.25) ÷ 10 = 6.65

### Step 4. Square root

The square root of 6.65 is 2.57876, which is your population standard deviation and your standard deviation for this set of values is 2.7183.

Below is a great explanation of how to interpret Standard Deviation

## What does standard deviation mean?

In the shortest explanation possible, it tells us the probability of a value occuring when given a data set (or set of values). For example, if the mean of a set of data is 50 and the standard deviation is 10, then there is a 68% probability that a number randomly picked from the set of values will be between 40 and 60. In this example, 1 standard deviation is 50 ±10, 2 standard deviations would be 50 ±20 (2 standard deviations have a 95% probability of occuring) and 3 standard deviations would be 50 ±30 (3 standard deviations have a 99.7% probability of occuring). Keep in mind this assumes it is a normal curve (bell curve).

A quick, hopefully practical example: You scored a 68% on an exam in school, when you first saw it you were petrified! Afterward, your professor said the average score on the exam was a 63%... okay, you did better than average. Then your professor says the standard deviation of the exam was 2%; now things are actually looking really good! If the class average was 63%, 1 standard deviation to the positive would be 65% (not to cause confusion here, but 68% of the class scored between 61-65%). 2 standard deviations means 95% of the class received a grade between 59-67%. Because you scored a 68%, you do not fall within the 95% of scores on the exam...the positive way to look at this is you scored better than 95% of the class.(hope this was a useful and relateable example).

## What are the uses for standard deviation?

There is tremendous value hidden in the data for those that collect data; even a motivated student might find it useful to determine the standard deviation of an exam to see how well they did in contrast to their peers. Standard deviation is utilized a lot in statistics and data analysis. The following is a list of examples where finding the standard deviation is useful:

- Marketing survey results

- Demographic data

- Forecasting

- Grades for a class

- Investing in the stock market