What Is Standard Deviation?
Standard Deviation (σ) measures how much individual data points vary from the mean. Standard deviation measures how spread out your data is. It applies in many fields. In finance, it helps analyze a portfolio of assets. In climate studies, it tracks temperature changes. It can also measure performance variation in games/sports. Standard deviation is important when working with expected value. It shows how much each value differs from the average.
How to Use the Standard Deviation Calculator?
Follow the below steps to calculate Standard Deviation using our standard deviation calculator
- Enter Your Data: Input your data set, separated by spaces, commas, or line breaks.
- Click "Calculate" to view standard deviation, variance, data count (n), mean, and sum of squares
- View the Calculation Steps: See the detailed steps of the calculation process.
- Copy and Paste: You can paste data lines directly from Excel or text documents, with or without commas. The table below shows acceptable formats.
Standard Deviation Formula:
1. Sample Standard Deviation:
The given formula is used for finding the standard deviation of a sample (subset of data drawn from the population):
\(s = \sqrt{\dfrac{1}{n - 1} \displaystyle\sum_{i=1}^n (x_i - \bar{x})^2}\)
Where
- s = Sample standard deviation
- \( x_{i}\) = Each single value in the data set
- x = Sample mean
- n = Total sample size
2. Population Standard Deviation:
When all the members of the population can be sampled, then the following standard deviation formula is used:
\(\sigma = \sqrt{\dfrac{1}{N} \displaystyle\sum_{i=1}^N (x_i - \mu)^2}\)
Where
- σ = Population standard deviation
- \( x_{i}\) = Individual value
- μ = Average mean value/expected value
- N = Total number of values
Other Statistical Formulas Used By Our Calculator:
The standard deviation calculator uses the following formulas to make statistical calculations of standard deviation:
Sum:
\(\text{Sum} = \displaystyle\sum_{i=1}^{n}x_i\)
Size, Count:
\(\text{Size} = n = \text{count}(x_i)_{i=1}^{n}\)
Mean:
For Sample:
\(\bar{x} = \dfrac{1}{n} \displaystyle\sum_{i=1}^n x_i\)
For Population:
\(\mu = \dfrac{1}{N} \displaystyle\sum_{i=1}^N x_i\)
Sum of Squares:
For Sample:
\(\ SS = \displaystyle\sum_{i=1}^{n} (x_i - \bar{x})^2\)
For Population:
\(\ SS = \displaystyle\sum_{i=1}^{N} (x_i - \mu)^2\)
Variance:
For Sample:
\(\ s^2 = \dfrac{\displaystyle\sum_{i=1}^{n} (x_i - \bar{x})^2}{n - 1}\)
For Population:
\(\sigma^{2} = \dfrac{\displaystyle\sum_{i=1}^{N}(x_i - \mu)^{2}}{N}\)
Sample vs. Population Standard Deviation
Check out the table below to clearly see the differences between sample and population standard deviation:
| Criterion |
Sample Standard Deviation (s) |
Population Standard Deviation (σ) |
| Formula |
\(s = \sqrt{\dfrac{1}{n – 1} \displaystyle\sum_{i=1}^n\left(x_{i} – \bar{x}\right)^2}\) |
\(σ = \sqrt{\dfrac{1}{N} \displaystyle\sum_{i=1}^N\left(x_{i} – μ\right)^2}\) |
| Use Case |
Used when only a subset of the total population is sampled |
Used when the entire population data is available |
| Example |
Analyzing test scores of 30 students in a class |
Analyzing test scores of all students in a school |
| Application |
Useful in studies, surveys, and research |
Useful in complete data analysis, such as census data |
| Bias Adjustment |
Divides by \(n - 1\) to correct bias |
Divides by \(N\), assuming all data points are known and included |
| Calculation |
Typically used when sampling data |
Used for calculating exact statistics from a full population |
How to Calculate Standard Deviation
-
Calculate the Mean:
Find the average of the data set by adding all values and dividing by the total number of values.
-
Find the Distance from the Mean:
Subtract the mean from each value in the data set.
-
Square Each Distance:
Square the result of each distance calculated in step 2.
-
Sum the Squared Values:
Add up all the squared distances.
-
Calculate Variance:
- For population variance: Divide the sum by the total number of values (N).
- For sample variance: Divide the sum by one less than the total number of values (n - 1).
-
Find the Standard Deviation:
Take the square root of the variance. The result is the standard deviation.
Example:
Suppose you have a data set (3, 4, 9, 7, 2, 5 ), find its standard deviation.
Solution:
Step #1(Calculate Mean Value):
\(\bar{x} = {\dfrac{3 + 4 + 9 + 7 + 2 + 5}{6}}\)
\(\bar{x} = \dfrac{30}{6}\) \(\bar{x} = 5\)
Step #2(Calculate The Value Of \(\left(x_{i} – \bar{x}\right)\):
|
Data Values (xi)
|
xi - x̅ |
(xi - x̅ )2 |
| 3 |
3 - 5 = -2 |
(-2)2 = 4
|
|
4
|
4 - 5 = -1 |
(-1)2 = 1 |
| 9 |
9 - 5 = 4 |
(4)2 = 16
|
|
7
|
7 - 5 = 2 |
(2)2 = 4 |
| 2 |
2 - 5= -3 |
(-3)2 = 9
|
|
5
|
5 - 5 = 0 |
(0)2 = 0
|
Step # 3 (Calculate Sample Standard Deviation):
Now we know that:
\(\ s = \sqrt{\dfrac{1}{n – 1} \displaystyle\sum_{i=1}^n\left(x_{i} – \bar{x}\right)^2}\)
\(\ s = \sqrt {\dfrac { 4+1+16+4+9+0}{ 6-1}}\)
\(\ s = \sqrt {\dfrac { 34 }{5}}\)
\(\ s = \sqrt {6.8}\)
Standard Deviation =\(\ s = 2.6076...\)
Corrected vs. Uncorrected Standard Deviation
For sample data, a correction is applied, known as the standard error of the mean. This ensures an unbiased estimation of the population standard deviation. This correction uses the formula (n-1) instead of n. It accounts for the fact that the sample only approximates the population. The adjusted calculation helps reduce bias in the estimate.
Standard Deviation, Variance, and the Bell Curve
In a normal distribution, shown as a bell curve, standard deviation is related to variance. Variance measures the squared deviations from the mean. Standard deviation is the square root of variance. Together, these two measures show how data points spread out from the expected value.
This is crucial when calculating confidence intervals in statistics. It is useful in estimating the reliability of your data. Confidence intervals show the range where the true population mean lies. It takes into account the standard error of the mean.
Real-World Applications
- Financial Analysis: Helps analyze financial portfolios.
- Sports Predictions: Aids in predicting outcomes in games/sports.
- Environmental Evaluation: Useful for evaluating environmental data.
- Informed Decisions: Understanding standard deviation helps you make better choices.
Tools Available: Use Excel spreadsheets and statistics formulas for easy calculations.
FAQs
Why is standard deviation important?
Standard deviation measures how much individual data points differ from the mean. It shows the spread of the data and helps you understand variability.
- Finance: Used in analyzing portfolios and assessing stock market risk.
- Climate Studies: Helps evaluate temperature fluctuations.
- Quantifies Uncertainty: Indicates the level of uncertainty or volatility in data.
- Confidence Intervals: Essential for determining the reliability of your data.
What’s the difference between sample and population standard deviation?
The difference depends on the dataset:
- Population Standard Deviation:
- Uses the entire dataset.
- Divide the sum of squares by the total number of data points (n).
- Sample Standard Deviation:
- Used when working with a sample of the population.
- Divides by (n-1) instead of n to correct for bias.
- This adjustment provides an unbiased estimate of the population's true standard deviation. It is called the corrected sample standard deviation.
Is a high or low standard deviation better?
It depends on the context:
- Stable Portfolio: A low standard deviation is better. It means less risk and lower price fluctuations.
- Games/Sports: A higher standard deviation can indicate desirable variability for competition.
- Stock Market: A higher standard deviation can indicate more chances for high returns. However, it also means there is greater risk involved.
References:
-
Frost, Jim. "Standard Deviation: Interpretations and Calculations." Statistics by Jim, 22 Feb. 2023. https://statisticsbyjim.com/basics/standard-deviation/.
-
"Standard Deviation." Wikipedia, Wikimedia Foundation, 15 Sept. 2023. https://en.wikipedia.org/wiki/Standard_deviation.
-
"Standard Deviation | Mean, Variance & Distribution." Britannica, Encyclopaedia Britannica, Inc., 15 Sept. 2023. https://www.britannica.com/topic/standard-deviation-statistics.
-
"Standard Deviation Formula and Uses vs. Variance." Investopedia, 5 Aug. 2024. https://www.investopedia.com/terms/s/standarddeviation.asp.
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