A statistical method called standard deviation is used to measure how variable or dispersed the data under observation are. It offers a crucial understanding of how the data points are distributed around the mean or average value.
In mathematics, the standard deviation is typically represented by the tiny alphabet, or Greek letter, sigma σ, which stands for the population standard deviation. This article will explain the idea of SD, explain its importance, and provide examples.
Definition: Standard Deviation
The assessment of variability is the simplest definition of Standard Deviation (SD) in statistical research. It is a phrase that is frequently and widely used in data analysis. It displays the extent of departure from the mean (average) of the data.
Depending on two potential scenarios, a low standard deviation (SD) indicates that the data points tend to be close to the mean, whereas a high SD indicates that the data are scattered throughout a large range of values.
Types of Standard Deviation:
As we have discussed the process of taking standard deviation, has two types.
- Population standard deviation
- Sample standard deviation
Population standard deviation:
A population’s level of variance among its individual data points is measured by the population standard deviation. To put it simply, it’s a means of measuring how dispersed the data is from its mean. When the numbers you have represent the entire population, it is relevant.
σ = √ {∑(xi – µ)2 / N}
Sample standard deviation:
A sample standard deviation is a statistic that is to be found out from only a few individuals in a prescribed population. It pointed towards the standard deviation of the sample rather than that of a population.
s = √ {∑(xi – x̄)2 / (N – 1)}
Both the sample and population STD formulas play a vital role in calculating standard deviation of the given data.
Applications of Standard Deviation:
In a variety of contexts, including those related to academics, business, finance, forecasting, manufacturing, the medical field, polls, and demographic features, among others, standard deviation helps us to understand the diversity in data collection.
It also teaches us how to utilize various methods for calculating the coefficient of variation, testing hypotheses, and calculating confidence intervals.
-
In Statistical Studies:
We have to acknowledge about coefficient of variation, hypothesis testing, confidence intervals, etc. When drawing conclusions from statistics and evaluating hypotheses, the standard deviation is crucial.
It aids in calculating the standard error, which is needed to run hypothesis tests and determine confidence intervals. For evaluating the accuracy and dependability of estimators and defining the statistical significance of findings, the standard deviation is crucial.
- In Manufacturing:
We have to know about quality control, improvement logic, and precision machining of parts to ensure proper size. Manufacturers can find causes of variability and identify the biggest contributors by computing the standard deviation for several elements.
It helps in monitoring and maintaining product quality standards.
- In Forecast Accuracy:
We have to know about weather conditions and natural disasters prevailing around due to climate change. In the context of weather forecasting, some related concepts and measures help us to assess uncertainty and variability in weather predictions.
These include ensemble forecasting, standard deviation of model output, and probabilistic forecasts.
- In Medicine:
In the medical field, there is always a chance to provide a drug that is more suitable and effective in a positive way. To improve the reliability of suitable drugs or medical equipment, this term provides a helping hand to test data repeatedly.
Through this technique, the most suitable product is provided in the market.
Examples
- In the prominent area of Punjab, a cellular company publishes the usage of cellular data in GB (Gigabytes) of 30 houses are given as below 73,75,58,57,86,76,47,66,53,52,43.
Compute the sample standard deviation of the given data.
Solution:
Step 1: N (number of terms) = 10
Step 2: ∑X= 73 + 75 + 58 + 56 + 76 + 47 + 66 + 53 + 52 + 43
∑X = 599
Step 3: Squared Difference:
∑(Xi – )2 = 1303.6
| No. of terms | Xi | Mean = X̅ | Xi – X̅ | (Xi – X̅)2 |
| 1 | 73 | — | 13.2 | 174.24 |
| 2 | 75 | — | 15.2 | 231.04 |
| 3 | 58 | — | -1.79 | 3.24 |
| 4 | 56 | — | -3.79 | 14.44 |
| 5 | 76 | — | 16.20 | 262.44 |
| 6 | 46 | — | -13.79 | 190.44 |
| 7 | 66 | — | 6.20 | 38.44 |
| 8 | 53 | — | -6.79 | 46.24 |
| 9 | 52 | — | -7.79 | 60.84 |
| 10 | 43 | — | -16.79 | 282.24 |
| ∑Xi = 599 | X̅ = 59.9 | — | ∑(Xi – X̅)2 = 1303.6 |
Step 4: Using the formula for sample data, we have
SD = √[∑(Xi – X̅)2 / N – 1]
SD = √(1303.6/9)
SD= √144.84
STDEV = s ≈ 12.036 (Answer)
- Prof. Jayson’s class got the marks as 98, 78, 72, 73, 59, 65, 83, 85, 80. Find the population standard deviation of the class scoring marks.
Solution:
| No. of Students | Xi | Mean = μ | Xi – μ | (Xi – μ)2 |
| 1 | 98 | — | 21 | 441 |
| 2 | 78 | — | 1 | 1 |
| 3 | 72 | — | -5 | 25 |
| 4 | 73 | — | -4 | 16 |
| 5 | 59 | — | -18 | 324 |
| 6 | 65 | — | -12 | 144 |
| 7 | 83 | — | 6 | 36 |
| 8 | 85 | — | 8 | 64 |
| 9 | 80 | — | 3 | 9 |
| ∑Xi = 693 | μ = 77 | — | ∑(Xi – μ)2 = 1060 |
Using the Population Standard Deviation formula,
SD = σ = √ (∑(xi – µ)2 / N)
σ = √ (1060 / 9)
σ = √ 117.78
STDEV = σ ≈ 10.853 (Answer)
Conclusion
In this article, we’ve explored the concept of STD, including its types, formulas, and solved examples. The understanding of its application in everyday life highlight its significance.
