Standard deviation serves as a statistical measure that gauges the dispersion of a set of numbers from their mean. Essentially, it provides valuable insights into how spread out the values are within a given dataset. The calculation of standard deviation involves determining the square root of the variance, with the variance computed by assessing the variation between each data point and the mean.
This approach gives more weight to outliers, preventing differences above the mean from canceling out those below and thereby avoiding a variance of zero. In this article, we will discuss the concept of Standard deviation of group data only, its measuring steps with formulas, and examples in different dimensions.
Understanding Standard Deviation
Standard deviation is defined as the statistical assessment of data spread or dispersion. It means that it shows how many variations are there from the mean, or to what extent the values typically deviate from the mean or average of the data set.
The interpretation of standard deviation lies in its ability to convey the degree of deviation within the dataset. When data points are situated farther from the mean, indicating a broader spread, the standard deviation is higher.
Conversely, when points are closer to the mean, showing a narrower spread, the standard deviation is lower. In essence, the standard deviation acts as a measure of the dispersion, revealing the extent to which numbers deviate from the central tendency of the data
Typically, it is symbolized as SD and denoted by the Greek word sigma σ
For Group data
σ = √[(∑FX2/∑F) – (∑FX/∑F)2]
For Ungroup data
σ = √[(∑X2/n) – (∑X/n)2]
Key Features of Standard Deviation
Standard deviation proves highly beneficial in the realm of statistics for the analysis of provided data. Examining its properties can provide a simplified approach to addressing various statistical problems.
1) The standard deviation for any constant is consistently zero. For any constant” a”, σ (a) = 0
2) The origin does not affect the standard deviation. σ (x ± a) = σ (x)
3) When all the values are multiplied by a constant, the standard deviation will be multiplied by that constant.
σ (ax) = a. σ (x) And σ (x/a) = (1/a) . σ (x)
4) The standard deviation of the sum or difference of two independent variables is equal to the sum of their respective standard deviation.
σ ( x + y ) = σ (x) + S.D. (y) And σ ( x – y ) = σ (x) + σ (y)
Calculating Steps for Standard deviation of group data
There are some steps for calculating the Standard deviation of group data. As in group data, classes and frequency are only two measures that are provided, so we have to calculate other metrics.
- Figure out midpoints of classes and write them in 3rd column by adding upper-class boundary and lower-class boundary and dividing that sum by 2.
- Take squares of each midpoint and write them in the 4th column in correspondence to each of the following.
- Multiply each frequency with its relevant midpoint value and write all these calculations in the 5th
- Multiply each value of frequency with respect to its squared value and write all the computations in the 6th and last column.
- Take the sum of the frequency column, 5th column, and 6th
- Substitute all values in the formula to evaluate the output of Standard deviation.
Calculating Steps for Standard deviation of Ungroup data
Below are a few steps to find standard deviation of ungrouped data.
- Add all the numbers to get ∑X, as far as its formula is concerned.
- Square all the numbers and add them to get ∑X2.
- Substitute the provided values into the given formula.
- Divide both calculations by the number of values “n”, as far as the formula is concerned.
Example
Example 1: (Group Data)
Find the Standard deviation for the following data:
| Range | 1-10 | 11-20 | 21-30 | 31-40 | 41-50 |
| Frequency | 2 | 7 | 10 | 3 | 1 |
Solution:
| Range | Frequency | X | X2 | FX | FX2 |
| 1-10 | 2 | 5.5 | 30.25 | 11 | 60.5 |
| 11-20 | 7 | 15.5 | 240.25 | 108.5 | 1681.75 |
| 21-30 | 10 | 25.5 | 650.25 | 255 | 6502.5 |
| 31-40 | 3 | 35.5 | 1260.25 | 106.5 | 3780.75 |
| 41-50 | 1 | 45.5 | 2070.25 | 45.5 | 2070.25 |
| ∑F = 23 | ∑FX = 526.5 | ∑FX2 =14095.75 |
Using the formula for group data, we have
σ = √[(∑FX2/∑F) – (∑FX/∑F)2]
Substituting values
σ = √ [14095.75/23 – (526.5/23)2]
σ = √612.86 – 524.01
σ = √88.85
⇒ σ = 9.43
Example 2: (Ungroup data)
The usage of electricity of specific area in winter is given as 63, 52, 32, 87, 36, 23, 75, 72, 83, 23, 78, 97, 31, 43. Calculate its Standard Deviation.
Solution:
Standard Deviation = √[(∑X2/n) – (∑X/n)2]
Step 1:
∑ (X) = (63 + 52 + 32 + 87 + 36 + 23 + 75 + 72 + 83 + 23 + 78 + 97 + 31+ 43)
∑ (X) = 795
∑ (X2) = (632 + 522 + 322 + 872 + 362 + 232 + 752 + 722 + 832 + 232 + 782 + 972 + 312+ 432)
∑ (X2) = 53621
Step 2:
Standard Deviation = √ [(53621/14) – (795/14)2]
= √ [ 3830.07 – 3224.61]
= √605.46
S.D. = 24.61
Wrap Up
In the above discussion, we explored the concept of standard deviation, its diverse formulations with respective formulas, and the properties in the analysis of standard deviation data. The usage of standard deviation becomes apparent as it finds application in various contexts through different illustrative examples. Exploring its practical applications in daily life serves as a constant reminder of its significance and relevance.
