In statistics, the standard deviation is created in statistics which denotes the sprinkling of data around its mean. The standard deviation is normally used in different statistical analyses such as hypothesis testing and confidence interval calculations. Also used in the field of finance to calculate the profit or volatility of a stock or investment forum.
The basic technique of standard deviation is thoroughly connected to other statistical calculations such as variance, which is the square of the standard deviation. The use of standard deviation extended to every field with economics, biology, and engineering.
The given data points are shown to be more closely centered on the mean if the standard deviation value is low and are more widely spread if the standard deviation value is very high.
In this article, we will discuss the basic definition of standard deviation, mathematical representation, working criteria of standard deviation, and its uses in detail. Moreover, example and their solution related to standard deviation are discussed here.
What is Standard deviation?
Large and less variability is the indicator of standard deviation, greater variability indicates a larger deviation. Understanding the dispersion of data points in a distribution, determining the level of risk or uncertainty in a dataset, and basing judgments on the consistency or variability of the data are frequently employed in statistics.
Standard deviation is a basic idea in statistics and plays an important role in fields such as science, finance, and quality control for process improvement.
Formula:
The formula for standard deviation is given below.
Procedure For Evaluating Standard Deviation
The main procedure to evaluate the standard deviation is as follows:
- To get started, determine the dataset’s mean (average).
- Add them all square value.
- Subtract one from the total amount of data points to get the sum of squared disparities.
- To find the standard deviation we take the square root on this step.
- Alternatively, you can use a shortcut formula by subtracting the mean squared from the mean of the squared values.
- Another way we use to find standard deviation we find the first variance and then we take the square root to find the standard deviation.
- When working with a population rather than an example, reduce the entire amount of observations by the sum of squared differences without deducting one.
- For grouped data, use the midpoints of each interval to calculate the mean and follow the same steps as above.
Uses of standard deviation
The standard deviation has different important uses in statistics and data analysis. Here are some common uses of the standard deviation,
The measure of variability:
The measure of variability in statistics provides information about the spread or dispersion of values within a dataset. Less variability represents a smaller standard deviation; a higher standard deviation signifies greater variability.
Descriptive statistics:
The standard deviation is a fundamental descriptive statistic that summarizes the dispersion of data. It complements measures of central tendency such as the mean or median by providing information about the distribution’s spread. Descriptive statistics, including the standard deviation help in summarizing and understanding data concisely.
Hypothesis testing:
The standard deviation plays a vital role in hypothesis testing where researchers compare observed data to expected values. It helps determine the significance of observed differences and evaluate the strength of evidence against a null hypothesis. For instance, in t-tests or ANOVA (analysis of variance), the standard deviation is utilized to calculate test statistics and assess the significance of differences between groups.
How to calculate problems of finding std?
You can calculate the standard deviation for a set of data either with the help of online tools like Standarddeviationcalculator.io or manually. Below are a few examples to help you learn how to solve problems manually.
Example 1:
Determine the standard deviation of the succeeding data sets 5, 8, 10, 13, and 22.
Solution:
Step 1:
Write the given data carefully.
dataset = {5, 8, 10, 13, 24},
N = 5
Standard deviation =?
Step 2:
Write the formula of Standard Deviation.
S. D = σ = √ [∑ (xi – x̄)2/ (N-1)]
Step 3:
Finding the mean of a given dataset.
x̄ = ∑ (xi)/N
x̄ = ∑ (5 + 8 + 10 + 13 + 24) /5
x̄ = 60/5
x̄ = 12
Step 4:
Evaluate the squared differences from the mean to evaluate the variance
⇒ (5 – 12)2 = (-7)2 = 49
⇒ (8 – 12)2 = (-4)2 = 16
⇒ (10 – 12)2 = (- 2)2 = 4
⇒ (13 – 12)2 = (1)2 = 1
⇒ (24 – 12)2 = (12)2 = 144
Step 5: Variance = (49 + 16 + 4 + 1 + 144) / (5 – 1)
V = Variance = 214 / 4
V = Variance = 53.55
Step 6:
For standard deviation take the square root of variance.
S.D = √53.5 = 7.31
S.D = 7.31
Example 2:
Given 1,3,5,7 sample data find the standard deviation.
Solution:
Step 1:
| No | xi | (xi – x̄) | (xi – x̄)2 |
| 1 | 1 | 1 – 4 = -3 | (-3)2 = 9 |
| 2 | 3 | 3 – 4 = -1 | (-1)2 = 1 |
| 3 | 5 | 5 – 4 = 1 | (-1)2 = 1 |
| 4 | 7 | 7 – 4 = 3 | (3)2 = 9 |
| n = 4 | x̄ = Σ(xi)/n = 16/4 = 4 | Σ (xi – x̄) = 0 | Σ (xi – x̄)2 = 20 |
Step 2:
Total terms = n = 4
Mean = x̄ = 4
Step 3:
Standard deviation formula for sample
= √Σ (xi – x̄)2/ (n – 1)
Step 4:
s² = Σ (xi – x̄)2/ (n – 1) = 6.66
Step 5:
s = √Σ (xᵢ – x̄)2/ (n – 1)
= √6.66 =
⇒s = 2.58
Summary
In this article, we have explored the standard deviation with definition, its formula, the working principle of standard deviation, and its uses in detail. Additionally, the standard deviation is explained with the help of examples in detail. Everyone reads this article and understands related problems.
FAQs
Q1:
Using the definition, discuss the idea of standard deviation.
Answer:
Standard deviation is a statistical measure of the dispersion or spread of data points in a dataset.
Q2:
Is it possible the population of standard deviation is zero?
Solution:
Yes, if all of the population’s data values are the same because there cannot be variation or expansion, the mean deviation of the population can be zero.
