The standard deviation is one of the common ways in statistics to measure the dispersion and variation in statistical data. It informs us how much each data point in a set is spread around the mean. A high standard deviation generally shows that the data points are more spread out, whereas a low standard deviation predicts that the data points are closer to the mean. Standard deviation is commonly used in statistics and economics to measure risk and variability.
The common abbreviation used for standard deviation is “SD” in mathematics. The lowercase Greek letter “σ” (sigma) represents population standard deviation, and the Latin letter “s” denotes sample standard deviation. A population refers to the entire data set under consideration, whereas a sample is a portion or subset of the population.
In this article, we will discuss the definition of standard deviation and its formula. We will learn how to determine the standard deviation through various examples.
In descriptive statistics, the standard deviation denotes the degree of dispersion or scatter within a dataset. It measures the average distance of each data point from the mean and tells how widely the data is spread out from the mean value. A higher standard deviation suggests that the data points are widely dispersed around the mean, while a smaller standard deviation expresses that most values in the dataset are near the mean.
Mathematically, Standard Deviation (SD) can be defined as the positive square root of the average of the squared deviation of the observation from their arithmetic mean. It helps in understanding the variation of data points relative to the average.
The standard deviation formula enables us to determine the amount of dispersion or scatter within a dataset. Different formulas are used to find the standard deviation based on whether the dataset is from the entire population or a sample.
If the data has been collected from every member of the population, then the formula to calculate the standard deviation is as follows:
If the data has been collected from a sample instead of every member of the population, then use the following formula to calculate the standard deviation.
Follow the following procedure to compute the standard deviation
Let’s consider some examples to understand the above step clearly.
Here are some examples on standard deviation with their step-by-step solution.
Example 1. (For Population)
Find the standard deviation for the given dataset. 11, 19, 23, 15, 27
Solution:
Step 1: Find the Population Mean (μ) of the given dataset.
μ = Sum of all observations in population / Number of observation
μ = 11, 19, 23, 15, 27 / 5 = 19
Step 2: Determine the difference between that observation and the population mean for each observation in the dataset. Also, take the Square of each of these differences.
| x | (x – μ) | (x – μ)2 |
| 11 | 11 – 19 = – 8 | (- 8)2 = 64 |
| 19 | 19 – 19 = 0 | (0)2 = 0 |
| 23 | 23 – 19 = 4 | (4)2 = 16 |
| 15 | 15 – 19 = – 4 | (- 4)2 = 16 |
| 27 | 27- 19 = 8 | (8)2 = 64 |
| å(x – μ)2 = 160 |
Step 3: Find the average of all the squared differences.
(å (x – μ) 2) / N) = 160 / 5
(å (x – μ) 2) / N) = 32
Step 4: Now, take the positive square root of this average.
σ = √ [å (x – x̄) 2 / N] = √32
σ = 5.66
A population standard deviation calculator can also be used to evaluate population STD of the given data with steps in order to avoid time-taking calculations.
Example 2. (For Sample)
Determine the standard deviation for the dataset below.
4, 9, 7, 15, 10
Solution:
Step 1: Calculate the sample Mean (x̄).
Sample means = x̄ = Sum of all observations / Number of observation
x̄ = 4, 9, 7, 15, 10 / 5 = 9
Step 2: Find the difference between that observation and the mean for each observation in the dataset. Also, take the Square of each of these differences.
| x | (x – x̄) | (x – x̄)2 |
| 4 | 4 – 9 = – 5 | (- 5)2 = 25 |
| 9 | 9 – 9 = 0 | (0)2 = 0 |
| 7 | 7 – 9 = – 2 | (- 2)2 = 4 |
| 15 | 15 – 9 = -6 | (- 6)2 = 36 |
| 10 | 10 – 9 = 1 | (1)2 = 1 |
| å(x – x̄)2 = 66 |
Step 3: Divide the sum of the squared differences by the sample size minus one (n – 1).
(å (x – x̄) 2) / (n – 1) = 66 / 4
(å (x – x̄) 2) / (n – 1) = 16.5
Step 4: Take the positive square root of the obtained result to get the sample standard deviation.
s = √ [å (x – x̄) 2 / (n – 1)] = √16.5
s = 4.06
In this article, we learned that standard deviation is a measure of dispersion or scatter from the average. We have discussed the formulas of standard deviation for both population and sample data. We covered the steps of calculating standard deviation that are easy to understand and provided examples to clarify these steps. After understanding this article, you will be able to find the standard deviation confidently.
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