Basics of Standard Deviation: Definition, Formula and Examples
Standard deviation gives a clearer understanding of the variability of data points around the mean value as well as summarizes the overall behavior of the data. It also helps to determine trends and patterns in datasets. The abbreviation “SD” is often used to represent the standard deviation.
It is also indicated by the Greek letter sigma (σ) for the population standard deviation and the Latin letter (s) for the sample standard deviation.
In this article, we will learn more about standard deviation and formulas to find. We will cover different properties of standard deviation that help us in solving the question. We will understand how to find SD through many examples.
Just stay with us to become an expert in calculating Standard deviation.
What is Standard Deviation (SD) in the Statistics?
In statistics, a standard deviation is a measure of the dispersion or variation in given data. Standard deviation provides information on how much individual data points differ from the mean of the dataset. It calculates the degree of spread within a group of numbers.
- Smaller standard deviations indicate that data points tend to be closer to the mean.
- Larger standard deviations show more variations in values.
Standard Deviation (SD) can be defined mathematically as the positive square root of the average of the squared differences between each data point and the mean. Standard deviation helps in understanding how much individual data point differs from the average value, which provides insight into the spread or dispersion within the dataset.
Standard deviation Formulas for Population & Sample
The standard deviation formula allows us to find the numerical amount of dispersion or scatter within a dataset. The calculation of standard deviation involves using different formulas based on whether the dataset represents the entire population or a sample.
The formulas for standard deviation are as follows:
Standard Deviation of Population Data
The population standard deviation formula is applied when we have data for the entire group or population and we want to measure the variability or spread of values around the population mean.
Population Standard Deviation (σ) Formula
σ = √ [Σ (xi – µ) 2 / N]
or
Shortcut methods = σ = √ [(Σ x 2 / N) – ([Σ x / N) 2]
Where:
- σ = standard deviation for population
- Σ = sum of …
- xi = each value in the population
- µ = the population mean
- N = the population size
Sample Standard Deviation
The sample standard deviation formula is used when we are working with a subset of data rather than the entire population and we want to measure the variability or spread of values around the sample mean.
Sample Standard Deviation (s) Formula
s = √ [Σ (xi – x̄) ² / (n – 1)]
Where:
- s = SD for the sample
- xi = each value in the sample
- x̄ = sample mean
- n = sample size
Note: The major difference between the population and sample standard deviation formulas is that for the sample, we divide by (n-1) instead of N, to provide an unbiased estimator of the population standard deviation.
Common Properties of Standard Deviation
There are several properties and methods associated with calculating the standard deviation:
- The standard deviation is always greater than or equal to zero (i.e. SD ≥ 0).
- The standard deviation will be zero if all of the data points in the set are the same.
- Adding or subtracting a constant from each value in the dataset does not affect its standard deviation.
- The standard deviation can be affected if there are any extreme values in the data set.
- The standard deviation is very helpful in identifying the shape of a distribution such as whether it is bell-shaped, skewed, or uniform.
How to Find Standard Deviation of Population and Sample?
Here are the steps to find the SD for both population and sample data:
- Determine the mean (average) of the given data set. (∴ Average = Sum of dataset / Number of items in the dataset).
- Determine the deviation of each observation from the mean of the dataset and take a square of each deviation.
- Sum all of the squared values obtained from step 2.
- Divide the sum of squared values by the total number of data points minus 1 for the population standard deviation. For sample data, divide the answer of result step 2 by the total number of data points.
- Now, take the positive square root to get SD.
Let us consider some examples to clarify these steps further.
Solved Examples
Example 1:
The distances covered by a runner in consecutive days are 3, 4, 5, 6, 7, 8. Calculate the standard deviation of the distances covered.
Solution:
Step One: Find the mean (average) of the distances:
Mean (or average) = x̄ = (3 + 4 + 5 + 6 + 7 + 8) / 6
= 33 / 6
x̄ = 5.5
Step Two: Determine the differences between each distance and the mean and take a square of each difference.
| x | (xi – x̄) | (xi – x̄)2 |
| 3 | 3 – 5.5 = -2.5 | (-2.5)2 = 6.25 |
| 4 | 4 – 5.5 = -1.5 | (-1.5)2 = 2.25 |
| 5 | 5 – 5.5 = -0.5 | (-0.5)2 = 0.25 |
| 6 | 6 – 5.5 = 0.5 | (0.5)2 = 0.25 |
| 7 | 7 – 5.5 = 1.5 | (1.5)2 = 2.25 |
| 8 | 8 – 5.5 = 2.5 | (2.5)2 = 6.25 |
Step Three: Find the sum of the squared differences.
Σ (xi – x̄)2 = 6.25 + 2.25 + 0.25 + 0.25 + 2.25 + 6.25 = 17.5
Step Four: Divide the sum by the total number of distances minus 1 (since it is a sample).
[Sum of (xi – x̄) ² / (n – 1)] = 17.5 / (6 – 1)
= 17.5 / 5
= 3.5
Step Five: Calculate the positive square root of the obtained result to find the standard deviation for the sample.
Standard deviation = √ [Σ (xi – x̄) ² / (n – 1)]
= √ (3.5)
= 1.87
Thus, the standard deviation of the distances covered by the runner over consecutive days is approximately 1.87 units.
The problems of standard deviation can also be solved by using a standard deviation formula calculator to quickly find the results with steps in order to save time.
Example 2.
Compute the standard deviation for the following dataset.
8, 8, 8, 8, 8, 8
Solution:
Standard deviation = 0
Because we know that if all of the data points in the set are the same, then the standard deviation will be zero.
Conclusion
In this article, we have explored the concept of standard deviation in depth. We examined different formulas for calculating the standard deviation for both sample and population data. We discussed properties that play a vital role in solving problems.
We provided steps to find the standard deviation (SD), and to understand them practically, we solved examples in the examples section. After reading this article, you will be able to calculate SD quickly and with ease.
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