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Standard Deviation

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 Sub Topics In statistics, standard deviation is a concept that is used frequently in the various fields. Standard deviation was suggested by Karl Pearson, a measure of dispersion in 1893. A measure of how spread out the numbers are from its mean. It is defined as the positive square root of the arithmetic mean of the squares of the deviations of the given observations from their arithmetic mean. Usually calculated as the square root of variance.

Formula for Standard Deviation

Population Standard Deviation :

If X$_{1}$, X$_{2}$, ..........................., X$_{n}$ is a set of n observations then its standard deviation is given by

$\sigma$ = $\sqrt{\frac{1}{N}\sum (X_{i} - \bar{X})^{2}}$

$\sigma$ : Population standard deviation

$\bar{X}$ : Population mean

X$_{i}$ : ith element from the population

N : Number of elements in the population.

Sample Standard Deviation :

If x$_{1}$, x$_{2}$, ..........................., x$_{n}$ is a set of n observations then its standard deviation is given by

s = $\sqrt{\frac{1}{n -1 }\sum (X_{i} - \bar{X})^{2}}$

s : Sample standard deviation

$\bar{X}$ : Sample mean

X$_{i}$ : ith element from the sample

n - 1 : Number of elements in the sample.

Standard deviation of the sample is an unbiased estimate of the standard deviation of the population.

How to do Standard Deviation?

Given below are the very basic steps to be followed for calculating standard deviation:

1) Compute the arithmetic mean $\bar{X}$
$\bar{X}$ = $\frac{1}{n}$ $\sum$ $X$

2) Compute the deviation (X - $\bar{X}$) of each observation from arithmetic mean.
(X$_{1}$ - $\bar{X}$), (X$_{2}$ - $\bar{X}$), ..........., (X$_{n}$ - $\bar{X}$)

3) Square each of the deviations obtained in the above step.

4) Calculate the sum of the squared deviations.
$\sum$ (X - $\bar{X}$)$^{2}$ = (X$_{1}$ - $\bar{X}$)$^{2}$ +  (X$_{2}$ - $\bar{X}$)$^{2}$ + ............+ (X$_{n}$ - $\bar{X}$)$^{2}$

5) Divide result obtained in step 4 by N or n - 1 depending on the problem.

6) Take the positive square root of the value obtained above.

7) Resulting value gives the standard deviation of the distribution.

Symbol for Standard Deviation

Standard deviation of a population is different from the standard deviation of a sample. Standard deviation of a sample and standard deviation of a population have different notations.
Standard deviation of a population is denoted by the Greek alphabet $\sigma$
Standard deviation of a sample is denoted by 's'.
If the value of standard deviation is found to be lower then the values will be very much closer to their average.
If the value of standard deviation is found to be higher then the values are scattered far from the average value.
They are generally used to group a normally distributed data set.

Standard Deviation Rule

The 68 - 95 - 99.7 rule is known as the three sigma rule or the empirical rule in statistics, where in all values lie within three standard deviations of the mean in a  normal distribution.
This rule is valid to a variable X having normal distribution with mean 'mu' and standard deviation 'sigma' and is not valid for distributions that are not normal.

Approximately 68.27% of the values lie within one standard deviation of the mean:

68.27% of the observations are between (mu - sigma) and (mu + sigma).
In mathematical notation it is written as,
P($\mu$ - $\sigma$ $\leq$ x $\leq$ $\mu$ + $\sigma$) $\sim$ 0.6827
where x : observation from a normally distributed random variable
$\mu$ : Mean
$\sigma$ : Standard deviation

Approximately 95.45% of the values lie within two standard deviation of the mean:

95.45%  of the observations are between (mu - sigma) and (mu + sigma).
In mathematical notation, P($\mu$ - $\sigma$ $\leq$ x $\leq$ $\mu$ + $\sigma$) $\sim$ 0.9545

Approximately 95.45% of the values lie within two standard deviation of the mean:

99.73%  of the observations are between (mu - sigma) and (mu + sigma).
In mathematical notation, P($\mu$ - $\sigma$ $\leq$ x $\leq$ $\mu$ + $\sigma$) $\sim$ 0.9973

Sample Standard Deviation

Sample standard deviation is an estimate based on sample, of a population standard deviation. It provides an important measure of variation or spread in a data set and is denoted by the small letter 's'.  Usually quite often your data is only a sample of the whole population. You may need to show how their sample results can be generalized to the whole population.

Standard Deviation Squared

Standard deviation is the square root of variance, so variance is the standard deviation squared.
Variance of a data set is the average of the squared differences from the mean. It is always a positive number and is a measure of how far each set is from the mean. Usually first variance is found then the standard deviation is found by squaring it.
It is also a measure of dispersion and is denoted by the greek letter $\sigma ^{2}$.

Formula for finding variance is:
$\sigma ^{2}$ = $\frac{\sum (x-\bar{x})^{2}}{n-1}$
where,
$x$ : Value of each observation.
$\bar{x}$ : Mean
n : Number of values
$\sum$ : Sum across the values

Variance V/S Standard Deviation

The table below clearly differentiates between variance and standard deviation:
 Variance Standard Deviation Expressed in square units Standard deviation is expressed in the same units of the original variable and is much easier to work with and easier to interpret in conjunction with the concept of the normal curve. Variance is very useful mathematically as the sum of uncorrelated distributions also has a variance that is the sum of the variances of those distributions. Describes the variability of the data.

Binomial Standard Deviation

One of the most commonly used statistical distribution is the binomial distribution.
It is a discrete probability distribution used to model the number of successes in a sample of size n. It is drawn from a population of size N and the draws are with replacement.
Binomial distribution is also known as bernoulli distribution and its probability mass function is:
$p(r)$ = P(X  = r) = $\binom{n}{r}p^{r}q^{n - r}$, r = 0, 1, 2, .....
where,
n = Total number of trials
r = Total number of successful trials
p = Probability of success in a single trial
q = Probability of failure in a single trial = 1 - p

Standard deviation of binomial distribution is:
$\sigma$ = $\sqrt{np(1 - p)}$

Uniform Distribution Standard Deviation

A distribution that has constant probability, and is also known as rectangular distribution.
In a discrete uniform distribution finite number of outcomes are equally likely to happen. It is inherently non parametric.

General formula for the probability density function of the uniform distribution is:

$f(x)$ = $\frac{1}{b - a}$ for a $\leq$ x $\leq$ b
a : location parameter
(b - a) : scale parameter

Equation for the standard uniform distribution is:
$f(x)$ = 1  for 0 $\leq$ x $\leq$ 1

The formula for standard deviation of uniform distribution is
$\frac{(b - a)}{\sqrt{12}}$

Percent Relative Standard Deviation

Relative standard deviation is the absolute value of the coefficient of variation. Expressed in terms of percentage and is abbreviated as RSD or %RSD.
It gives precision as a percentage of the mean.
Formula for relative standard deviation is :

%RSD = $\frac{Stdev}{Mean}$  $\times$ 100%

Should be written with two significant figures.
A lower percentage indicates lower variability and a higher variability indicates the data is more varied.

Standard Deviation Examples

Example 1 : For the given below data set find the standard deviation.
10, 2, 38, 23, 38, 23, 21

Solution : Total number of terms, n = 7

$\bar{x}$ = 22.1428

Formula for finding the sample standard deviation is

s = $\sqrt{\frac{1}{n -1 }\sum (X_{i} - \bar{X})^{2}}$

s = 13.2844
Therefore for the given data set the standard deviation is found to be 13.2844

Example 2 : For the given below data set find the standard deviation.
155, 142, 142, 214, 212, 242, 254, 235, 252, 145

Solution :

Here n = 10 (Total number of terms)

$\bar{x}$ = 199.3

Formula for finding the sample standard deviation is

s = $\sqrt{\frac{1}{n -1 }\sum (X_{i} - \bar{X})^{2}}$

s = 47.9839
Therefore for the given data set the standard deviation is found to be 47.9839.