Standard deviation is a measure of data dispersion derived by taking the square root of variance and is a cornerstone of statistical analysis. It is shortened as SD and represented by the symbol “σ” and it indicates the extent to which data values differ from the average value (mean). A low standard deviation implies that the values are more closely clustered around the mean. A high standard deviation indicates a significant deviation from the mean value.
This article will cover the concept of SD, its definition, formula, how we calculate, and methods to calculate ungrouped data. We’ll also some examples to gain more insight into the SD.
What is the SD?
Standard deviation (SD) is a statistical measure that quantifies the dispersion or spread of data points within a dataset from its average (mean) and provides insight into the variability of the dataset. To shed more light on this, let us consider an example.
Example:
Imagine measuring the heights of a small group of people to understand how Standard Deviation (SD) works. The SD reveals how spread out or clustered our height measurements are around the average height. To calculate the mean, we sum up all the measured heights and divide the total by the number of individuals. Let’s assume our mean height is 155cm.
Now, let’s examine how each person’s height differs from this mean. For instance, the first person might be 18cm taller than the mean, the second person 8cm taller, and so on. This illustrates that individual significantly taller or shorter than the mean contributes to larger deviations.
However, our interest lies in understanding how much, on average, these individuals deviate from the mean height. In essence, the standard deviation precisely tells us this average deviation from the mean. In our example, the standard deviation calculates to 12.06cm, indicating that, on average, people in our group deviate around 12.06cm from the mean height of 155cm. The above figure helps us grasp the overall variability in height within this particular group.
SD Formula:
Data sets can be categorized into populations and samples. A population refers to the complete set of individuals under study, while a sample represents a smaller subset selected from the larger population for analysis. The formulas for standard deviation vary slightly when dealing with a population compared to when working with a sample.
Here are both formulas to calculate SD in sample and population:
The process to calculate the SD:
The calculation of SD involves multiple steps. We can grasp how SD is calculated by following this sequential procedure.
- Calculate the mean (average) of the dataset.
- Find the difference between each data point and the mean.
- Square these differences.
- Find the average of these squared differences.
- Take the square root of this average to get the standard deviation.
How do we calculate the SD of Ungrouped data?
The computation of standard deviation varies depending on the nature of the data. It quantifies how data deviates from its mean or central position. Approaches to calculate standard deviation involve three distinct methods.
- Actual Mean Method
- Assumed Mean Method
- Step Deviation Method
- Actual Mean Method:
The “Standard Deviation by The Actual Mean Method” refers to the process of calculating standard deviation using the mean (average) of the dataset. This method involves finding by the following formula:
- Assumed Mean Method:
The “Standard Deviation by Assumed Mean Method” is a statistical approach used to calculate the standard deviation of a dataset by first assuming a mean (average) value. In this method, the variance is computed by determining the squared differences between each data point and the assumed mean.
- Step Deviation Method:
The Step Deviation Method is applicable not only for ungrouped data but also for calculating the standard deviation of grouped data. In this method, an arbitrary data value, denoted as A, is selected as the assumed mean. Subsequently, the deviations of all data values are determined by subtracting this assumed mean (‘A’) from each data point (x) to get ‘d = x – A’.
Following this, step deviations (d prime) are computed using d divided by a common factor i, where i is a shared factor among all d values. The formula used for calculating the standard deviation of grouped data by the step deviation method is then applied.
Problems of Standard Deviation:
Problem 1:
Calculate the standard deviation for a sample data set consisting of values:
| x | 12 | 15 | 18 | 21 | 24. |
Solution:
Step 1: Find the mean of the dataset
x̄ = (12 + 15 + 18 + 21 + 24) / 5 = 18
Step 2: Find the deviation of each data point from the mean
Step 3: Square each deviation
| X | X – x̄ | (X – x̄)2 |
| 12 | -6 | 36 |
| 15 | -3 | 9 |
| 18 | 0 | 0 |
| 21 | 3 | 9 |
| 24 | 6 | 36 |
| – | – | ∑ (Xi – x̄)2 = 90 |
Step 4: Find the average of these squared deviations
x̄ = (36 + 9 + 0 + 9 + 36) / 5 = 18
Step 5: Take the square root of the average:
s = √18 ≈ 4.24
The standard deviation for this sample data set is approximately 4.24.
You can also use an SD calculator to verify your results or finding solutions with steps to avoid manual calculations.
Final Words:
In this article, we explored Standard Deviation (SD), a vital statistical measure displaying data variability around the mean. We delved into its definition, formulas for sample and population, and methods to calculate SD for ungrouped data. We covered the Actual Mean, Assumed Mean, and Step Deviation methods with clear examples. By solving problems, we illustrated how to compute SD for datasets.
