Understanding Standard Deviation: Definition, Formulas, Steps, and Examples
Standard deviation is a fundamental statistical concept that measures the degree of dispersion or variability within a dataset. It offers insightful information about how certain data points differ from the mean or average of the dataset.
While a big standard deviation denotes a substantial spread, suggesting increased variability, when data are closely packed around the means this type of standard deviation means a small standard deviation. Understanding standard deviation is essential for making informed decisions, analyzing risk, and drawing meaningful conclusions in various fields, from finance and science to education and healthcare.
This article will cover the standard deviation idea, the procedures for calculating it, the formulas of standard deviation, and the importance of standard deviation.
Definition of standard deviation
In statistics, the standard deviation is a measure of the average deviation or spread of a function or data set from its mean value, often used in the context of probability distributions and statistical analysis.
Formulas used for finding standard deviation
There are different methods and formulas for calculating standard deviation, depending on the context and available data.
| Population Standard Deviation (σ): | σ = √[∑(xi – μ)2/N] | σ denotes the standard deviation of the population.∑ is the sum symbol.xi represents individual data points.μ is the population mean.N = all given numbers. |
| Standard Deviation of Sample (s): | s = √[∑(xi – x̄)2/n – 1] | Sample size of all data points denotes s.∑ is a summation symbol.xi represents individual data points.x̄ is the sample mean.n is the sample size. |
Steps for Finding Standard Deviation
Here are the steps to find standard deviation of population and sample set of data.
Step 1: Collect the data we want to find the standard deviation
First of all, we select those data where we want to find the standard deviation This data could represent anything from test scores and financial returns to temperature readings and more.
Step 2: Determine all given data mean
The formula used for finding means
μ (means) = (∑xi) / N
- μ represents the mean.
- ∑xi = all points of data sum.
- N = all given data number.
Step 3: Deviations finding
The deviation (d) for each data point is:
d = xi – μ
- Single deviation represents d.
- xi = individual data points.
- μ = means values.
Step 4: Take all Deviations square
In this step, we take a square of the step 3 answer value. Squaring the deviations ensures that both positive and negative deviations contribute to the measure of variability.
The squared deviations (d²) are:
d² = (xi – μ) ²
Step 5: Variance calculation
The formula for the variance (σ²) is:
σ² = (Σ (xi – μ) ²) / N
- σ² represents the variance.
- ∑ (xi – μ) ² denotes the sum of squared deviations.
- N = all given data points.
Step 6: Evaluate our required standard deviation
To get the standard deviation, take the variance value obtained in step 5 and square root (√) it.
σ = √σ²
You can also use online tools like Standarddeviationcalculator.io to find the standard deviation of the given sample or population data according to the above-discussed steps.
The Importance of Standard Deviation in Different Fields
Standard deviation is used in different fields but here we discuss the sum of popular fields.
- Science and Research: Scientists use standard deviation to analyze experimental data and determine the reliability of their results. It is crucial for concluding fields like physics, biology, and chemistry.
- Quality Control: In manufacturing and quality control, standard deviation is used to monitor and maintain the consistency and quality of products.
- Education: In education, standard deviation helps evaluate the performance of students and assess the effectiveness of teaching methods.
- Economics: Economists use standard deviation to analyze economic data and assess the stability and variability of economic indicators.
- Healthcare: In healthcare, standard deviation is used to analyze patient data, clinical trials, and medical research, helping doctors and researchers make informed decisions.
Examples of Standard Deviation
Example 1: Let’s suppose the sample space data 11, 3, 40, 15, 10 and 11.
The standard deviation of sample space =?
Solution: Given data of sample space
X = 11, 3, 40, 15, 10 and 11
Step 1: Find the μ (mean) of the given data sample space
x̄ = (11+ 3 + 40 + 15 + 10 + 11) / 7
x̄ = 110/ 7
x̄ = 15.7143
Step 2: Calculate the deviation then take the square and add all of them
Deviation = xi – x̄
| Data values (xi) | xi – x̄ | (xi – x̄)2 |
| 11 | 11 – 15.7143 | (11 – 15.7143)2 |
| 3 | 3 – 15.7143 | (3 – 15.7143 )2 |
| 40 | 40 – 15.7143 | (40 – 15.7143)2 |
| 15 | 15 – 15.7143 | (15 – 15.7143)2 |
| 10 | 10 – 15.7143 | (10 – 15.7143)2 |
| 11 | 11 -15.7143 | (11 -15.7143)2 |
After simplification, we get
∑ (xi – x̄)2 = 847.4284
Step 3: The formula used for sample space of standard deviation
s = √[∑(xi – x̄)2/n – 1]
Step 4: Put the required value in the given formula of sample space we get the required sample space standard deviation.
s = √ [1/ (7-1)] (847.4284)
s = √ (0.1667) (847.4284)
s = √ 141.2381
s = 11.88
Example 2: Find the population standard deviation of 6, 14, 35, 18 and 30
Solution
Step 1: Find the means of given data
Means = total value sum/5
Mean = 103/ 5
Mean = 20.6
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 |
| 6 | 6 – 20.6 = -14.6 | (-14.6)2 = 213.16 |
| 14 | 14 – 20.6 = -6.6 | (-6.6 )2 = 43.56 |
| 35 | 35 – 20.6 = 14.4 | (14.4)2 = 207.36 |
| 18 | 18 – 20.6 = -2.6 | (-2.6)2 = 6.76 |
| 30 | 30 – 20.6 = 9.4 | (9.4)2 = 88.36 |
Step 3:
Now used formula of standard deviation for the population
σ = √[∑(xi – μ)2/N]
Put all required values in the formula we get our required results
σ = √ (1/5) (213.16 +43.56+207.36+6.76+88.36)
σ = √ (0.2) (559.2)
σ = 10.58
Conclusion
In this article, we have discussed the concept of standard deviation, the step-by-step finding process, formulas, and the practical application of two’s standard deviation also discussed here, to enhance your understanding, we have provided examples throughout the discussion. I hope you will able to defend this article anywhere after completing studying this article.
Related article: How to Make Statistics Easier Using a Critical Value Calculator
FAQs
How do we separate the sample standard deviation and population?
Answer: When you have data from a sample of the population, you use the sample standard deviation, but the population standard deviation is used for data from the full population. The formulas for these two calculations differ slightly.
What other statistical measures work well with standard deviation?
Answer: Measures like mean, median, and quartiles complement standard deviation by providing a more comprehensive view of data. These measures help describe the central tendency and distribution of data.
How are manufacturing and quality control processes using standard deviation?
Answer: In quality control and manufacturing, standard deviation is used to monitor and maintain the consistency and quality of products. It helps identify variations in product specifications and process control.
