Standard Deviation is a measure of how data is spread out around the mean. It is a statistical tool used to determine the amount of variation or dispersion of a set of values from the mean. A low standard deviation indicates that the data points are clustered closely around the mean, while a high standard deviation means that the data points are spread across a wide range of values.

In this article, we will discuss how to calculate Standard Deviation using a formula.

What is Standard Deviation?

Standard deviation is a statistical measure that quantifies the amount of variation or dispersion in a set of data values. In simpler terms, it indicates how much the individual data points in a dataset deviate from the mean (average) of the data.

Standard deviation is a way to measure how spread out numbers are in a group. Imagine you have a set of numbers and you want to know how different these numbers are from each other. If the numbers are very similar to each other, the standard deviation will be small. But if the numbers are all over the place and very different, the standard deviation will be large.

Read More about Standard Deviation.

Formula for Standard Deviation

The formula for standard deviation depends on whether you are dealing with a full population or just a sample from that population. Thus, there are two formulas are:

• Population Standard Deviation
• Sample Standard Deviation

Let’s discuss these formula in detail as follows:

Population Standard Deviation

Formula for population standard deviation is given as follows:

σ = √Σ(xi – μ)² / N

Where,

• σ (sigma) is the population standard deviation
• Σ (sigma) is the summation symbol
• x_i is the ith value in the data set
• μ is the population mean
• N is the total number of values in the population

Sample Standard Deviation

Formula for sample standard deviation is given as follows:

s = √Σ(x_i – x̅)²/(N – 1)

Where,

• s is the sample standard deviation,
• Σ (sigma) is the summation symbol,
• x_i is the i^th value in the data set,
• x̅ is the sample mean, and
• N is the total number of values in the sample.

Steps for Calculations of Standard Deviation

We can calculate the standard deviation using following steps:

Step 1: Calculate the mean (average) of the data set.

Step 2: For each data point, subtract the mean and square the difference.

Step 3: Sum the squared differences.

Step 4: Divide the sum of squared differences by N – 1 (for sample standard deviation) or N (for population standard deviation).

Step 5: Take the square root of the result.

Example on Standard Deviation Calculation

Let’s calculate the standard deviation of the following data set: {2, 4, 5, 7, 9}

Step 1: Calculate the mean for the data

Mean = (2 + 4 + 5 + 7 + 9) / 5 = 5.4

Step 2: Calculate the squared deviations from the mean.

• (2 – 5.4)² = 11.56
• (4 – 5.4)² = 1.96
• (5 – 5.4)² = 0.16
• (7 – 5.4)² = 2.56
• (9 – 5.4)² = 13.69

Step 3: Sum the squared deviations from the mean.

Σ(xi – μ)² = 11.56 + 1.96 + 0.16 + 2.56 + 13.69 = 29.93

Step 4: Divide the sum of squared differences by N – 1.

29.93 / (5 – 1) = 5.99

Take the square root of the result.

s = √5.99 ≈ 2.45

Therefore, the sample standard deviation of the data set is approximately 2.45.

Read More,

• Mean in Statitics
• Variance
• Mean, Variance and Standard Deviation
• Mean Absolute Deviation
• Normal Distribution

Solved Examples of Standard Deviation

Example 1: Calculate the population standard deviation for the following data set: {6, 8, 10, 12, 14}

Solution:

Mean = (6 + 8 + 10 + 12 + 14) / 5 = 50 / 5 = 10

Now, calculate the squared deviations from the mean,

• (6 – 10)² = 16
• (8 – 10)² = 4
• (10 – 10)² = 0
• (12 – 10)² = 4
• (14 – 10)² = 16

Thus, Σ(xi – μ)² = 16 + 4 + 0 + 4 + 16 = 40

Divide the sum of squared differences by N – 1, and take square root to find standard deviation,

σ = √(40/5) = √8 ≈ 2.83

So, the population standard deviation of the data set is approximately 2.83.

Example 2: Calculate the sample standard deviation for the following data set: {12, 15, 18, 21, 24}

Solution:

Mean = (12 + 15 + 18 + 21 + 24) / 5 = 90 / 5 = 18

Now, calculate the squared deviations from the mean,

• (12 – 18)² = 36
• (15 – 18)² = 9
• (18 – 18)² = 0
• (21 – 18)² = 9
• (24 – 18)² = 36

Thus, Σ(xi – μ)² = 36 + 9 + 0 + 9 + 36 = 90

Divide the sum of squared differences by N – 1, and take square root to find standard deviation,

s = √(90/4) = √22.5 ≈ 4.74

Therefore, the sample standard deviation of the data set is approximately 4.74.

Practice Problems on Standard Deviation

Problem 1: Given the test scores of a class, 65, 70, 78, 72, 68, 74, 81, 70, calculate the standard deviation.

Problem 2: A farmer measures the weight of ten pumpkins in pounds: 12, 15, 17, 11, 16, 14, 15, 16, 14, 15. Compute the standard deviation to understand the variability in pumpkin weights.

Problem 3: Two teachers recorded the scores of their students on the same exam. Teacher A’s student scores: 88, 92, 76, 94, 85. Teacher B’s student scores: 85, 83, 84, 87, 86. Calculate and compare the standard deviation of scores from both classes.

Problem 4: Consider the ages of participants in a study: 34, 37, 29, 31, 38, 36, 30, 33. Calculate the standard deviation and discuss what this might suggest about the spread of ages in the study.

Problem 5: A basketball player’s points per game over ten games are recorded as follows: 22, 28, 26, 32, 24, 19, 35, 27, 23, 31. Find the standard deviation to evaluate the consistency of the player’s scoring.

Calculation of Standard Deviation: FAQs

Define Standard Deviation.

Standard deviation is a statistic that measures the dispersion or variability of a dataset relative to its mean.

How is Standard Deviation Calculated?

To calculate the standard deviation:

1. Find the Mean of the dataset,
2. Calculate the Variance,
3. Take the square root of the variance to find Standard Deviation.

What are the Types of Standard Deviation?

There are two main types:

• Population Standard Deviation: Used when an entire population can be measured and is denoted as σ (sigma).
• Sample Standard Deviation: Used when data represents a sample of a larger population, denoted as s.

Why Use (N-1) for Sample Standard Deviation?

Using N−1 instead of N (the number of observations) corrects the bias in the estimation of a population variance and standard deviation from a sample.

What Does a High Standard Deviation Indicate About Data?

A high standard deviation indicates that the data points are spread widely from the mean, suggesting a high level of variability in the data set. This can imply diverse data points and less consistency in the data set.

What Does a Low Standard Deviation Indicate About Data?

A low standard deviation indicates that the data points tend to be very close to the mean, suggesting low variability in the dataset. This typically implies that the data points are consistent and do not vary widely from the mean.

Can Standard Deviation Be Negative?

No, standard deviation cannot be negative because it is calculated as the square root of a variance, and the square root of a positive number or zero cannot be negative.