Introduction
One of the most basic approaches of statistical analysis is the standard deviation. The Standard Deviation, abbreviated as SD and represented by the letter ", indicates how far a value has varied from the mean value. A small Standard Deviation means the results are close to the mean, whereas a big Standard Deviation means the data are widely divergent from the mean. Let's look at how to determine the Standard Deviation of grouped and ungrouped data, as well as the random variable's Standard Deviation.
Standard Deviation
The Standard Deviation is a statistic that indicates how much variance or dispersion there is in a group of statistics. A low Standard Deviation means that the value is close to the mean of the set (also known as the expected value), and a high Standard Deviation means that the value is spread over a wider area.
The lower case Greek letter sigma, for the population Standard Deviation, or the Latin letter s, for the sample Standard Deviation, is most usually represented in mathematical texts and equations by the lower case Greek letter sigma.
The square root of the variance is the Standard Deviation of a random variable, sample, population, data collection, or probability distribution. It is algebraically easier than the average absolute deviation, but it is less resilient in practice. The Standard Deviation has the advantage of being reported in the same unit as the data, unlike the variance.
Standard Deviation is the measure of the dispersion of data from its mean. It measures the absolute variability of a distribution. The higher is the dispersion or variability of data, the larger will be the standard deviation and the larger will be the magnitude of the deviation of value from the mean whereas the lower is the dispersion or variability of data, the lower will be the standard deviation and the lower will be the magnitude of the deviation of value from the mean. The standard deviation formula is used to find the values of a specific data that is dispersed from the mean value. It is important to observe that the value of standard deviation can never be negative.
There Are Two Types of Standard Deviation
Population Standard Deviation
Sample Standard Deviation
Standard Deviation formula to calculate the value of standard deviation is given below:
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Standard Deviation Formulas For Both Sample and Population
Notations For the Sample Standard Deviation Formula and Population Standard Deviation Formula
σ = Standard Deviation
Xi = Terms given in the data
N = Total number of terms
\[\overline{X}\]= Mean of the data
What is Variance and Standard Deviation?
Variance - The variance is a numerical value that represents how broadly individuals in a group may change. The variance will be larger if the individual observations change largely from the group mean and vice versa.
It is important to notice similarities between the variance of sample and variance population. They have different representations and are calculated differently. The variance of a population is represented by σ² whereas the variance of a sample is represented by s².
Standard Deviation - Standard deviation is a measure of dispersion in statistics. It gives an estimation of how individuals in data are dispersed from the mean value. Standard deviation is defined as the square root of the mean of a square of the deviation of all the values of a series derived from the arithmetic mean. It is also known as root mean square deviation.The symbol used to represent standard deviation is Greek Letter sigma (σ 2).
Variance and Standard Deviation Formula
Variance, \[\sigma^{2} = \frac{\sum_{i=1}^{n} (x_{i} - \overline{x})^{2}}{n} \]
Standard Deviation, \[\sigma = \sqrt{\frac{\sum_{i=1}^{n} (x_{i} - \overline{x})^{2}}{n}} \]
In the above variance and standard deviation formula:
xi = Data set values
\[\overline{x}\]= Mean of the data
With the help of the variance and standard deviation formula given above, we can observe that variance is equal to the square of the standard deviation.
Mean and Standard Deviation Formula
The sample mean is the average and is calculated as the addition of all the observed outcomes from the sample divided by the total number of events. Sample mean is represented by the symbol. In Mathematical terms, sample mean formula is given as:
\[\overline{x} = \frac{1}{n} \sum\limits_{i=1}^{n} x \]
In the above sample mean formula
N is the sample size and
X is the correspond observed values
Standard Deviation - On the other hand, standard deviation perceives the significant amount of dispersion of observations when comes up close with data. In Mathematical terms, standard dev formula is given as:
Standard Deviation, \[\sigma = \sqrt{\frac{\sum_{i=1}^{n} (x_{i} - \overline{x})^{2}}{n}} \]
Standard Error of Mean Formula
The standard error of the mean is a procedure used to assess the standard deviation of a sampling distribution. It is also known as standard deviation of the mean and is represented as SEM. Generally, the population mean approximated value is the sample mean, in a sample space. But, if we select another sample from the same population, it may obtain a different value.
Therefore, a population of the sampled means will appear to have different variance and mean values. The standard error of the mean can be determined as the standard deviation of such a sample means including all the possible samples drawn from the same population. SEM is basically an approximation of standard deviation, which has been evaluated from the sample.
The standard error of the mean formula is equal to the ratio of the standard deviation to the root of the sample size.
\[SEM = \frac{SD}{\sqrt{N}} \]
In the above standard error of mean formula,
‘SD’ is the standard deviation
N is the number of observations.
Variance and Standard Deviation Formula for Grouped Data
\[\sigma = \frac{\sum f(m - \mu)^{2}}{N} \]
And
\[s^{2} = \frac{\sum f(m - \overline{x})^{2}}{n - 1} \]
The calculation of standard deviation can be done by taking the square root of the variance. Hence, the standard deviation is calculated as
Population Standard Deviation - \[\sigma = \sqrt{\sigma^{2}} \]
Sample Standard Deviation - \[s = \sqrt{s^{2}} \]
Here in the above variance and std deviation formula,
σ2 is the population variance, s2 is the sample variance, m is the midpoint of a class.
Standard Deviation Formula for Discrete Frequency Distribution
For the discrete frequency distribution of the type.
y : y1, y2, y3,y4
f : f1, f2, f3, f4
The formula for standard deviation becomes
\[ \sqrt{\frac{1}{N} \sum\limits_{i = 1}^{n} f_{i}(x_{i} - \overline{x})^2 }\]
In the above formula, N is the total number of observations.
Standard Deviation V/S Variance
The list of standard deviation v/s variance is given below in tabulated from
Solved Examples
1. During a survey, 6 students were asked the number hours per day they give time to their studies on an average? The answers of the students are as follows: 2, 6, 5, 3, 2, 3. Calculate the standard deviation.
Solution:
Step 1: Calculate the mean value of the given data
\[ = \frac{2 + 6 + 5 + 3 + 2 + 3}{6}\]
\[ = \frac{21}{6}\]
= 3.5
Step 2: Construct a table for the above given data
Step 3 : Now, use the standard dev formula.
Sample Standard Deviation Formula - \[s = \sqrt{\frac{\sum (x_{i} - \overline{x})^{2}}{n-1}} \]
\[= \sqrt{\frac{13.5}{6 - 1}}\]
\[= \sqrt{\frac{13.5}{5}}\] = \[= \sqrt{2.7}\]
= 1.643
2. A Hen lays eight eggs. The weight of each egg laid by hen is given below.
Solution:
Step 1: Let us first calculate the mean of the above data
Mean = \[\sum \frac{X}{N} \]
\[= \frac{60 + 56 + 61 + 68 + 51 + 53 + 69 + 54}{8} \]
\[= \frac{472}{8}\]
= 59
Step 2: Construct a table for the above - given data
Step 3 : Now, use the standard dev formula
Standard Deviation Formula \[= \sqrt{\frac{\sum (x_{i} - \overline{x})^{2}}{n}} \]
\[= \sqrt{\frac{320}{8}}\] = \[ \sqrt{40} \]
= 6.32 grams
Quiz Time
1. Find the Standard Deviation for the Given Data
5,10,7,12,10,20,15,22,8.2
6.89
10.01
7.26
9
2. Which of the Following Is the Measure of Variability?
Mean
Median
Mode
Standard Deviation
FAQs on Standard Deviation Formulas
1. What is a Standard Deviation?
In descriptive statistics, the standard deviation is the degree of dispersion or scatter of data points relative to the mean. It is a measure of the data points' deviation from the mean and describes how the values are distributed over the data sample. The standard deviation of a sample, statistical population, random variable, data collection, or probability distribution is the square root of the variance.
When we have a certain amount of observations and they are all different,
\[x_{1},x_{2},x_{3},x_{4},x_{5}............x_{n}\]
then the value's mean deviation from the mean is calculated as
\[\sum_{i=1}^{n} (x_{i} - \overline{x})^{2}\]
On the other hand, the sum of squares of deviations from the mean does not appear to be a reliable measure of dispersion. The observations are near to the mean when the average of the squared differences from the mean is low. This is a less dispersed level of dispersion. If this number is large, it implies that the observations are dispersed from the mean to a greater extent. As a result, we conclude that:
\[\sum_{i=1}^{n} (x_{i} - \overline{x})^{2}\]
is a good indicator of how dispersed or scattered something is.
2. What is the standard deviation formula?
The method of determining the deviation of a data point is used to calculate the degree of variance. Dispersion is discussed in summary statistics. The variance of a data set is the average square distance between the mean value and each data value, as previously stated. The standard deviation, on the other hand, is the range of data values around the mean.
Calculating Standard Deviation: A Step-by-Step Guide
Find the arithmetic mean of the observations, which is the mean.
Calculate the squared deviations from the mean. (Mean of the data value)2
Calculate the mean of the squared differences. (Variance = sum of squared differences multiplied by the number of observations.)
Calculate the square root of the variance. (Variance = Standard deviation)
3. What is Standard Deviation of Random Variables?
The degree to which the values depart from the predicted value is determined by the measure of spread for the probability distribution of a random variable. This is a function that gives each outcome in a sample space a numerical value. Because it is a function, it is indicated by X, Y, or Z. The standard deviation of a random variable is calculated by taking the square root of the product of the squared difference between the random variable, x, and the expected value () and the probability associated value of the random variable.
The probability distribution's standard deviation \[ X = x^{2}P(X = x) \]
4. What is Standard Deviation of Probability Distribution?
Many trials make up the experimental probability. We tend to know the average outcome when the difference between the theoretical probability of an event and its relative frequency approaches zero. The predicted value of the experiment, denoted by, is known as this mean.
The mean of a normal distribution is zero, while the standard deviation is one.
The number of successes is a random variable in a binomial experiment. The standard deviation of a random variable with a binomial distribution is: = npq, where mean: = np, n = number of trials, p = probability of success, and 1-p =q represents the probability of failure.
The standard deviation of a Poisson distribution is equal to t, where t is the average number of successes over a time span of t.
5. What is a variance?
Despite the fact that standard deviation is the most significant tool for measuring dispersion, it is critical to understand that it is generated from variance. Variance is better than mean deviation since it employs the square of deviations. However, because variance is based on squares, the square of the unit of items and means in the series is the unit of variance.
Statisticians use the square root of the variance, also known as standard deviation, to account for this. The standard deviation is effectively the square root of the variance. We can easily calculate variance as the square of standard deviation if we know how to calculate standard deviations.
Variance = ( Standard deviation)² = σ×σ
6. What are the Different Properties of Standard Deviation?
Some different properties of standard deviation are given below:
Standard deviation is used to compute spread or dispersion around the mean of a given set of data.
The value of standard deviation is always positive. It can never be negative.
Standard deviation is speedily affected outliers. A single outlier can increase the standard deviation value and in turn, misrepresent the picture of spread.
For data with almost the similar mean, the larger the spread, the greater the value of standard deviation.
If all values in a given set are similar, the value of standard deviation becomes zero (because each value is equivalent to the mean).
7. Mention Some Basic Points on Difference Between Standard Deviation and Variance?
The difference between standard deviation and variance is given below in tabulated form:
Variance | Standard Deviation |
Variance is simply stated as the numerical value, which mentions how variable in the observation are. | Standard deviation is simply stated as the observations that are measured through a given data set. |
Variance is nothing but average taken out from the standard deviation. | Standard deviation is stated as the root of the mean square deviation. |
It is defined using squared units | It is defined using the same units of the data available |
Mathematically, variance is denoted as (σ2) | Mathematically, variance is denoted as (σ) |
Variance is the accurate estimate of the individuals spread out in the group | Variance is the accurate estimate of the observations in a given data set. |
8. What is the Relative Standard Deviation?
Relative standard deviation is one of the measures of deviation of a set of numbers dispersed from the mean and is computed as the ratio of stand deviation to the mean for a set of numbers. Larger the deviation, further the numbers are dispersed away from the mean. Lower the deviation, the close the numbers are dispersed from the mean. It is also called a coefficient of variation.
The formula for the relative standard deviation is given as:
RSD = \[ \frac{s \times 100} {\text{X bar}}\]
In the above relative standard deviation formula.
RSD = Relative standard deviation
S =Standard deviation
X bar = Mean of the data