Introduction to Standard Deviation: Types and Examples
In the field of Statistics, the level of variability in under-observation data is to be measured in many statistical computations. For this purpose, the concept of Standard Deviation is used. It provides us with the guidelines to check how data points change for the mean or the average value. It’s a widely used metric for variability as it provides measurements in the same units as the original data.
Standard Deviation has two conditions. Firstly, a low standard deviation suggests that the dataset values are likely near the mean, while a high standard deviation shows that the dataset values are considerably distant from the mean.
In this article, we will discuss standard deviation along with its types, calculating steps for both types of data i.e. formula for group data and formula for ungroup data, and examples from different directions in detail.
Table of Contents
Definition of Standard Deviation.
The standard deviation is defined as the degree of dispersion of data under observation with respect to its mean value. It signifies how the dataset values are distributed within the sample and reflects the degree of variability of data points from the mean.
The SD of a data set, sample, or random variable is to be evaluated by finding the square root of variance. Standard Deviation is usually written as SD and symbolized by the Greek letter sigma (σ)
Types of Standard Deviation
- Population SD
The population standard deviation is a metric for determining how widely distributed the individual data points are within a population. It is a method to measure how dispersed the data is from the mean; to put it simply. When the data you have is representative of the entire population; its standard deviation is to be calculated by the formula given below.
σ = √ {(xi – µ) 2 / N}
2. Sample SD
A sample standard deviation is a statistical measure calculated from a sample of a larger population. It estimates the dispersion of a population based on a sample, providing an approximation of the population STD when the complete data set is not available.
The calculation of the sample standard deviation differs slightly from that of the population standard deviation to account for the smaller sample size, using the below formula.
S = √ [(xi – µ) 2 / (N – 1)]
Calculating Steps for Population Standard Deviation
Population Standard Deviation is calculated as follows:
- To start; the initial step involves determining the average of the provided data set. The mean is to be found by taking the sum of all numbers of the data set and dividing that sum by several values in the data set.
- Compute the variance of individual data points. The variance of data is to be calculated by subtracting the mean from the value of the data point.
- Squaring the variance of each data point.
- Sum of squared variance values.
- Divide the sum of squared variance values by the number of data points in the data set.
Determine the square root of the result obtained by dividing.
Calculating Steps for Sample Standard Deviation
The steps for calculating sample standard deviation are as follows:
- To start; the initial step involves determining the average of the provided data set. The mean is to be found by taking the sum of all numbers of the data set and dividing that sum by a number of values in the data set.
- Take a square of the value of the mean.
- Squaring all the numbers of the data set.
- Take a sum of squared numbers.
- Divide the sum of squared numbers by the total number.
- Subtract the value of the square of the mean from it.
- Finally, take the square root of the above value to find the output of Standard Deviation.
Examples
Example 1: (For Population Standard Deviation)
On a Test day of math with 15 points possible, the scores of eight test-takers were 7, 5, 9, 11, 10, 9, 12, 9.
Calculate the population SD.
Solution:
Step 1: Find the mean of the above data values
μ = ( 7 + 5 + 9 + 11 + 10 + 9 + 12 + 9 )/8 = 72/8 = 9
Step 2: The squared differences are
(7-9)2 = (-2)2 = 4
(5-9)2 = (-4)2 = 16
(9-9)2 = (0)2 = 0
(11-9)2 = (2)2 = 4
(10-9)2 = (1)2 = 1
(9-9)2 = (0)2 = 0
(12-9)2 = (3)2 = 9
(9-9)2 = (0)2 = 0
Step 3: The average of squared differences is
(4+16+0+4+1+0+9+0)/8 = 34/8 = 4.25
Step 4: The Standard Deviation is, therefore
σ = √4.25 = 2.06
σ = 2.06
Example 2: (For Sample Standard Deviation)
Compute S.D for the following data
23, 44, 59, 58, 87, 38, 78, 39, 24, 74, 76, 47, 34, 38, 66, 12, 73, 44
Solution:
Standard Deviation = √[(∑X2/n) – (∑X/n)2]
Step 1:
∑ (X) = 914
∑ X2 = 54330
Step 2:
Standard Deviation = √ [(54330/18) – (914/18)2]
= √ [ 3018.33 – 2577.59]
= √439.95
S.D. = 20.97
Wrap Up
In this section, we have discussed the concept of Standard deviation precisely in different directions. As it has been clarified to know the variation or measure of the dispersion of given data, we easily can find it through standard deviation.
Depending upon the nature of the data, two formulas are discussed above in this article. By understanding its examples, we can solve all statistical computations easily. Many of the basic properties of Standard deviation have also been mentioned above through which many complex or infinite data sets are analyzed easily.
FAQ's
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Any Algebra 1 student who wants to achieve an A grade must master the understanding of these concepts and abilities.
- Arithmetic
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The following fundamental ideas during Algebra 1.
- Simplifying
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FAQ's
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