Introduction to Standard Deviation: Explanation & Calculations
Standard deviation provides valuable insights into how spread out the data points are around the mean or average. In other words, standard deviation helps us understand the extent to which individual data points deviate from the average.
The concept of standard deviation is widely used in various fields, including mathematics, statistics, economics, finance, social sciences, and natural sciences. It plays a crucial role in data analysis, research, and decision-making processes.
In this article, we will discuss the definition of Standard Deviation, Method and formulas of Standard Deviation and interpretation of standard deviation.
Standard Deviation
The statistical metric of standard deviation quantifies the degree of variance or dispersion in a dataset, demonstrating how far the data points are from the mean. By calculating and interpreting standard deviation, we can gain a deeper understanding of the distribution of data and make informed conclusions about the reliability and consistency of the dataset.
It allows us to compare different datasets, identify patterns, detect outliers, and assess the risk or uncertainty associated with a particular set of observations.
Formulas of Standard Deviation
The standard deviation population formula is:
σ = √ (∑ (x – μ) ² / N)
Here,
- Standard deviation denotes σ.
- Sigma denotes ∑
- x shows each data point in the population,
- Population means is μ.
- N stands for the population’s total number of data points.
For the sample, the formula is different from the population formula.
s = √ (∑ (x – x̄) ² / (n – 1))
Here,
- s denotes the sample standard deviation,
- ∑ denotes the sum of,
- x is the data points,
- x̄ is sample mean, and
- n is total data points.
Steps to Find Standard Deviation
To calculate standard deviation, try to take assistance from online tools or follow the below steps for manual calculations:
- First of all, find the mean (average) of the dataset.
- For every data point, subtract the mean and then square the result.
- Find the mean of the squared differences obtained in step 2.
- And the last step takes a square root of the mean calculated in step 3.
There are slight variations in the calculation depending on whether you are working with a population or a sample. It depends on whether we are working with a population or sample space. Both are different formulas
By following these steps and using the appropriate formula, you can calculate the standard deviation for a given dataset.
Understanding of Standard Deviation
The interpretation of standard deviation is based on the magnitude of its value and provides insights into the dispersion or variability of a dataset. Here are three common interpretations:
- Low Standard Deviation:
The data points are thought to be densely packed around the mean if the standard deviation is low. The values in the dataset tend to be relatively similar and less spread out. This suggests that the data is more precise or consistent, with less variability or uncertainty.
- High Standard Deviation:
Data points are more widely dispersed from the mean when the standard deviation is high. The values in the dataset exhibit greater variability and may be farther from the average. This suggests that the data is more dispersed or less consistent, with more variability or uncertainty.
- Comparing Standard Deviations:
When comparing standard deviations between two or more datasets, the one with the larger standard deviation has greater variability and dispersion. It indicates that the values in that dataset are more spread out from the mean compared to the dataset with the smaller standard deviation. This comparison allows for understanding the relative differences in variability and dispersion among datasets.
How to find standard deviation?
Example 1: For sample standard deviation
Find the population standard deviation of 10, 2, 38, 18, 35 and 23
Solution
Step 1: Evaluate mean.
Mean of sample data = ∑x/n = [10+ 2+ 38+ 18+ 35+ 23]/6
= 126/6
= 21
Step 2: Now find the typical distance of each data point & the mean and square of each deviation
| Data values (xi) | xi – µ | (xi – µ)2 |
| 10 | 10 – 21 = -11 | (-11)2 = 121 |
| 2 | 2 – 21 = -20 | (-20)2 = 400 |
| 38 | 38 – 21 = 17 | (17)2 = 289 |
| 18 | 18 – 21 = 3 | (3)2 = 9 |
| 35 | 35 – 21 = 14 | (14)2 = 196 |
| 23 | 23 – 21= 2 | (2)2 = 4 |
Step 3: Add the deviations to find the statistical sum of squares.
∑ (xi – µ)2 = 121 + 400 + 289 + 9 + 196 + 4
∑ (xi – µ)2 = 1089
Step 4: Now divide the given sum of squares by n-1.
∑ (xi – µ)2/n = 1019/6
∑ (xi – µ)2/n-1 = 254.75
Step 5: Take the square root.
√ [∑ (xi – µ)2/n-1] = √254.75
√ [∑ (xi – µ)2/n-1] = 15.961
Example 2: For population standard deviation
Find the population standard deviation of 8, 4, 38, 18, 37 and 29
Solution
Step 1: Evaluate mean.
Mean of population data = ∑ x/n
= [8+ 4+38+ 18 +37 +29]/6
= 134/6
= 22.34
Step 2: Now find the typical distance of each data point & the mean and square of each deviation.
| Data values (xi) | xi – µ | (xi – µ)2 |
| 8 | 8 – 22.34 = -14.34 | (-14.34)2 = 205.64 |
| 4 | 4 – 22.34 = -18.34 | (-18.34 )2 = 336.35 |
| 38 | 38 – 22.34 = 15.67 | (15.67)2 = 245.55 |
| 18 | 18 – 22.34 = -4.34 | (-4.34)2 = 18.84 |
| 37 | 37 – 22.34 = 14.66 | (14.66)2 = 214.92 |
| 29 | 29 –22.34 = 7.34 | (7.34)2 = 53.87 |
Step 3: Add the deviations to find the statistical sum of squares.
∑ (xi – µ)2 = 205.64 + 336.35 + 245.55 + 18.84 + 214.92 + 53.87
∑ (xi – µ)2 = 1075.17
Step 4: Now divide the sum of squares by n.
∑ (xi – µ)2/n = 1075.17/6
∑ (xi – µ)2/n = 63.667
Step 5: Take the square root.
√ [∑ (xi – µ)2/n] = √63.667
√ [∑ (xi – µ)2/n] = 179.195
Frequently asked question
Question 1:
Reason of importance standard deviation.
Answer:
Standard deviation provides useful information about the spread or variability of data. Understanding how much each data point deviates from the average is useful. It is frequently used to analyze and contrast data sets in statistics, finance, and other domains.
Question 2:
What relationship does standard deviation have to variance?
Answer:
Standard deviation and variance are related to each other. The square variance response is the same as the standard deviation. The standard deviation is frequently selected since it is in the same units as the data, even though variance also gives a measure of dispersion.
Question 3:
How may the standard deviation be applied while making decisions?
Answer:
Standard deviation helps in decision making by providing insights into the level of uncertainty or risk associated with a set of data. It can be used to compare the variability of different data sets, assess the reliability of predictions or estimates, and make informed choices based on the degree of variability in the data.
Summary
In this article, we have discussed the definition of Standard Deviation, Method and formulas of Standard Deviation and interpretation of standard deviation. Also, with the help of example standard deviation will be explained. After studying this article anyone can defend this article easily.
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