Understanding Standard Deviation: Calculation Steps and Examples

Understanding Standard Deviation: Calculation Steps and Examples

Standard deviation is a way to measure how spread out or diverse the data is. It tells you how much the numbers in a group differ from the average. For instance, in a farming village where most people do similar jobs and earn similar incomes, the standard deviation would be small, showing less difference between earnings. But in a big city with diverse jobs and various income levels, the standard deviation would be higher, indicating more variation in earnings among people.

This article aims to shed light on the following noteworthy points of the Standard Deviation:

  • Definition of Standard Deviation.
  • How to Determine Standard Deviation?
  • Normal distribution in Standard Deviation
  • Empirical Rule

Let’s discuss all the above points and try to gain more understanding of the Standard deviation.

Standard Deviation Definition:

Standard deviation represents how much individual data points differ from the average and shows the extent of spread or variability within the dataset. A low SD suggests that data points are clustered close to the mean while a high SD indicates a wider distribution and greater variability.

Formula to computing the SD:

In SD, we use two types of formulas which are:

  1. Population in SD
  2. Sample in SD

Population Standard Deviation (σ):

  • Population: Refers to the entire group or set of individuals, items, or data points under study.
  • Population Standard Deviation (σ): It calculates how spread out the values are within the entire population.
Understanding Standard Deviation: Calculation Steps and Examples

Sample Standard Deviation (s):

  • Sample: A smaller group or subclass selected from the whole population.
  • Sample Standard Deviation (s): It calculates how spread out the values are within the sample.
Understanding Standard Deviation: Calculation Steps and Examples

The formula for sample standard deviation is quite similar but has a slight adjustment to account for the smaller size of the sample.

How to determine Standard Deviation?

Calculating standard deviation involves various steps:

Step 1: Calculate the Average: 

  • Add up all the values in your dataset and divide the sum by the total number of values. This gives you the mean.

Step 2: Find Differences from the Mean: 

  • Minus the mean from every value in the dataset.

Step 3: Square the Differences: 

  • Square each of these differences to make all values positive and emphasize the spread.

Step 5: Put Values in the Given formula:

  • Place the values in the given formula and take the answer

Normal Distribution in SD:

The relationship between standard deviation and normal distribution is crucial. When we visualize normal distribution on a graph, characterized by its bell-shaped curve, we’re effectively presenting how similar and independent data points are spread out. Within this distribution, data tends to arrange itself symmetrically around the mean in consistent intervals.

Now, considering standard deviation, the graph’s shape tells us a lot: a tall, narrow peak with minimal spreading implies a low standard deviation, while a flatter, wider curve signifies a higher standard deviation.

Understanding Standard Deviation: Calculation Steps and Examples

Understanding of Empirical Rule In SD:

We can apply the empirical rule for interpretation should our dataset follow a normal distribution pattern. This rule essentially indicates that nearly all the data points observed will lie within 3 SD from the mean value.

Here’s the breakdown:

  • 68% of the data points in a normal distribution fall within one SD of the mean. This is represented by the central bulge of the bell-shaped curve that characterizes a normal distribution.
  • 95% of the data points fall within two SD of the mean. This encompasses the central bulge and the gently sloping sides of the bell curve.
  • 99% of the data points fall within three SDs of the mean. This includes the entire bell curve and a tiny portion of the tails.
Understanding Standard Deviation: Calculation Steps and Examples

Solved Problems of Standard Deviation:

Problem 1:

Imagine a group of 10 students who participated in a mathematics examination. Their scores are:

Scores78828572906875887981

Solution:

Step 1: Calculate the mean score.

Mean = 78 + 82 + 85 + 72 + 90 + 68 + 75 + 88 + 79 + 81 / 10 = 798 / 10 = 79.8

Step 2:  Find the differences from the mean for each score.

XX – x̄
78-1.7999
822.2000
855.2000
72-7.7999
9010.2000
68-11.7999
75-4.7999
888.2000
79-0.7999
811.2000

Step 3: Square The differences

(X – x̄)2
3.24
4.84
27.04
60.84
104.04
139.24
23.04
67.24
0.64
1.44
∑ (Xi – X)2 = 431.6

Step 4:  Submit values in the formula  

S (SD)= 6.925

Problem 2: Annual Rainfall

Consider the annual rainfall (in inches) in a region over ten years:

Years45485042556043475358.

Solution:

  1. Calculate the mean score:

= 50.1

  • Find the differences from the mean for each value.
XX – x̄
45-5.100
48-2.100
50-0.100
55-8.100
604.899
439.899
47-7.100
53-3.100
582.899
  • Square these differences.
(X – x̄)2
26.01
4.41
0.01
65.61
24.01
98.01
50.41
9.61
8.41
62.41
∑(Xi – X)2 = 348.900
  • Calculate the average of these squared differences (variance).

Variance = 38.767

  • Take the Sqrt of the variance to find the SD.

S (SD) = 6.226

Wrap Up:

In this article, we explored the concept of standard deviation. It reveals how much values differ from the mean. We discussed its definition, calculation methods for both population and sample data, and its relationship with normal distribution and the empirical rule. We demonstrated how to compute standard deviation step by step by solving problems.

Leave a Reply

Your email address will not be published. Required fields are marked *