## How to Find Standard DeviationThe ## Standard Deviation DefinitionsThe standard deviation (SD) is a quantification that measures the distribution (dispersion) of the data set relative to its mean. It is calculated as the square root of the variance. It is denoted by the lower Greek letter Some other definitions are: - The standard deviation is the
**measure of the variations**of all values from the mean. - Standard deviation is the square root of the sum of squared deviation from the mean divided by the number of observations.
- It is the square root of the
**variance**.
## VarianceIt defines how much a random variable differs from its expected value. It is the average of the squares of the differences between expected and individual value. It can never have a negative value. It is denoted by σ ## How to Find Standard DeviationIt is calculated as the square root of variance by determining the variations between each data point relative to the mean. The higher the standard deviation, the higher the variance between each data set, and the mean. ## The formula of Standard DeviationThere are two formulas to calculate the standard deviation. Both formulas measure the variations. But there is a difference between them. - Population Standard Deviation
- Sample Standard Deviation
## Population Standard DeviationIt is a parameter that calculates a fixed value from every individual in the population. The formula of population standard deviation is: Where:
## Sample Standard DeviationIt is a statistic. In this standard deviation, only some individuals from the population are taken for the calculation. It has greater variability because it depends on the sample. Hence, the standard deviation of the sample is greater than the population standard deviation. The formula of sample standard deviation is: Where:
Now, we will see how these standard deviations are different from each other. Consider the sample and population standard deviation formula; we see that both the formulas are nearly identical.
- While using the
**population standard deviation**, divide the sum of the squared deviation by**N**(number of elements or observations). - While using
**sample standard deviation**, divide the sum of the squared deviation by**N-1**(one less than the number of elements or observations).
The values of population and sample standard deviation depend on ## Properties of Standard Deviation- The value of standard deviation is never negative.
- The low deviation indicated that the data point tends to very closer to the mean.
- High deviation indicates that the data point is spread out over a large range of values.
- If we add a constant to all the data sets, it does not affect the standard deviation.
- If we multiply a constant to all the data set, it affects the standard deviation.
- The standard deviation can be zero if and only if all the observations have the same value.
## Uses of Standard Deviation- It is widely used in biological studies, statistics, and financial field.
- It is used in fitting a normal curve to a frequency distribution.
- It is used to measure of dispersion.
- It is also used in the finance field to calculate financial risk.
## Methods of Standard Deviations## Direct MethodWe can also find the standard deviation by using the direct method. It is used when the deviation is taken from the actual mean. The formula for the direct method is: Where:
## Assumed Mean MethodIn this method, we do not calculate the actual mean. Instead of this, we choose a random value to calculate the deviation. The assumed value must lie around the middle value. It is also known as the Where,
## Step Deviation MethodIt is an extended form of the shortcut method. It simplifies the calculation. The formula for the assumed mean method is: Where,
## Types of DistributionsBefore moving to the examples, we must know about the three types of distribution. **Individual Series:**Individual series is a single column observation. For example:
**Discrete Series:**In discrete series, there are two columns. The first column consists of the observations, and the second column consists of the frequencies. For example:
**Frequency Distribution:**Frequency distribution also has two columns. The first column consists of the observations, and the second column consists of the frequencies. The observations are further classified into the intervals called classes. For example:
## Example of Individual Series
First, we will calculate the mean. Now, we will calculate the variance The formula for variance is:
Putting the values in the variance formula, we get: The formula for standard deviation is: σ =√σ σ =√310.625=17.624
We know the formula of the assumed mean method for individual series: In the above formula,
Putting the values in the above formula, we get: ## Example of Discrete Series
First, we will calculate the mean. We know the formula of a direct method for discrete series:
Putting the values in the formula, we get:
We know the formula of the shortcut method for discrete series: In the above formula,
Putting the values in the formula, we get:
## Example of Frequency Distribution (Grouped Data or Continuous Series)
We know the formula of step deviation method for continuous series: In the above formula, . Where A is assumed mean. So, first, we will calculate the mean (m). In the following table, we have calculated the mean of each class interval. Among them, we have assumed a mean that is
Putting the values in the formula, we get: ## Example of Population Standard Deviation
In the above question, the marks of ten students are given. The question says that apply the sample standard deviation. In this case, we will not take all student's marks for calculation. We will take a few student's marks for calculation as a sample. We have taken only
We know the formula of sample standard deviation: Now, we will find the values used in the formula.
N-1. Here, a total of 6 elements are there, so dividing the sum by 6-1=5, we get:
s =√209.2=14.46
## Example of Sample Standard Deviation
In the above question, the marks of ten students are given. The question says that apply the population standard deviation. In this case, we will take all student's marks for calculation. We know the formula of sample standard deviation: Now, we will find the values used in the formula.
N. Here, a total of 10 elements are there, so dividing the sum by 10, we get:
σ =√401=20.02=20
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