Standard deviation is a measure of how dispersed data values are from the mean. It is calculated as the square root of the variance, which is the average of the squared differences from the mean. A lower standard deviation means values are closer to the mean, while a higher standard deviation means values are more spread out from the mean. The document provides steps to calculate standard deviation for both grouped and ungrouped data, as well as formulas for finding the standard deviation of a sample or entire population.

1. Standard deviation?

2. Standard Deviation
1. Standard deviation is the positive square root of the variance.
2. Standard deviation is one of the basic methods of statistical analysis.
3. Standard deviation is commonly abbreviated as SD and denoted by 'σ’ and it
tells about the value that how much it has deviated from the mean value.
4. If we get a low standard deviation then it means that the values tend to be
close to the mean whereas a high standard deviation tells us that the values
are far from the mean value.

3. Standard Deviation
5. Let us learn to calculate the standard deviation of grouped and ungrouped
data and the standard deviation of a random variable.

4. What is Standard Deviation?
1. Standard deviation is the degree of dispersion or the scatter of the data points
relative to its mean, in descriptive statistics.
2. It tells how the values are spread across the data sample and it is the measure
of the variation of the data points from the mean.
3. The standard deviation of a sample, statistical population, random variable,
data set, or probability distribution is the square root of its variance.

5. 3. When we have n number of observations and the observations are x1,x2,.....xn,
then the mean deviation of the value from the mean is determined as
∑ni=1(xi−¯x)2.
4. However, the sum of squares of deviations from the mean doesn't seem to be a
proper measure of dispersion.
5. If the average of the squared differences from the mean is small, it indicates
that the observations xi are close to the mean ¯x.

6. 6. This is a lower degree of dispersion. If this sum is large, it indicates that there
is a higher degree of dispersion of the observations from the mean ¯x. Thus we
conclude that ∑ni=1(xi−¯x)2 is a reasonable indicator of the degree of dispersion
or scatter.

8. We take 1/n∑n
i=1(xi−¯x)2 as a proper measure of dispersion and this is
called the variance(σ2). The square root of the variance is the standard
deviation.

9. Steps to Calculate Standard Deviation
1. Find the mean, which is the arithmetic mean of the observations.
2. Find the squared differences from the mean. (The data value - mean)2
3. Find the average of the squared differences.
(Variance = The sum of squared differences ÷ the number of
observations)
4. Find the square root of variance. (Standard deviation = √Variance)

10. Standard Deviation Formula
1. The spread of statistical data is measured by the standard deviation.
2. The degree of dispersion is computed by the method of estimating the
deviation of data points.
3. You can read about dispersion in summary statistics.
4. As discussed, the variance of the data set is the average square distance
between the mean value and each data value.
5. And standard deviation defines the spread of data values around the
mean.

11. Here are two standard deviation formulas that are used to find the standard
deviation of sample data and the standard deviation of the given
population.

13. Ungrouped data

18. Discrete data

22. Continuous data