Standard Deviation: Explanation and Calculations of Sample and Population STD
Standard Deviation is a statistical terminology used to find
or measure the amount of scatterness around an average of under-observation
data. Dispersion or scatterness is the difference between the original and
average values. The more maximize this dispersion or variability is, the more bigly
the value of the standard deviation.
Standard Deviation may be represented by S.D and is
most commonly symbolized in mathematics by the small alphabet (a Greek letter)
sigma Ïƒ, for the
population S.D.
The standard Deviation of a random variable or sample is also
calculated by taking the square root of its variance. Finding it by taking the square
root is simpler algebraically.
Both the standard deviation of the data and the standard error of the
estimate are frequently discussed (as a measure of possible mistakes in the
findings).
What is the Standard Deviation?
In statistical investigations, the measurement of variability
known as the standard deviation (SD) is often and extensively employed. It
reveals the degree of deviation from the data's average (mean). A low SD
suggests that the data points tend to be near the mean, whereas a high SD
suggests that the data are dispersed throughout a wide range of values,
depending on two possible situations.
Types
of Standard
Deviation:
As we have
discussed that the process of taking Standard deviation, it has two types.
· Population standard deviation
· Sample standard deviation
Population Standard
Deviation:
How much variance there is
between specific data points in a population is shown by the population
standard deviation. We may sum it up by saying that it is a means to measure
how dispersed the data is from the mean. When the numbers you have are for the
entire population, it is relevant.
Ïƒ = √ {(xi - Âµ)^{ 2} / N}
Sample Standard Deviation:
A sample standard deviation
is a statistical measure that is computed from only a
few values in a reference population. It pointed towards the standard deviation of the
sample rather than that of a population.
S = √ {(xi - Âµ)^{ 2} / (N – 1)}
Examples of Sample
and Population STD
Example 1: (For population
standard deviation)
Four
colleagues were comparing their scores on a recent test.
Calculate the standard deviation of their scores: 5,7,6,2
Solution:
Step 1: Find the mean.
Mean
(Âµ) = 5+7+6+2/4
= 20/4
= 5
The
mean is 5 points.
Step 2: Each
score will subtract the mean of the data.
Score (x_{i}) |
Deviation (x_{i
} _{− }Âµ_{ }) |
5 |
5 -_{ }5 = 0 |
7 |
7 - 5 = 2 |
6 |
6 - 5 = 1 |
2 |
2 - 5= - 3 |
Step 3: Square
each deviation.
Score (x_{i}) |
Deviation (x_{i
} _{− }Âµ_{ }) |
(x_{i } _{− }Âµ_{ })^{2} |
5 |
5 -_{ }5 = 0 |
0 |
7 |
7 - 5 = 2 |
4 |
6 |
6 - 5 = 1 |
1 |
2 |
2 - 5= - 3 |
9 |
Step 4: Add
the squared deviations.
0+4+1+9=14
Step 5: Division by number of scores.
=√14/4
=3.5
Step 6: To
Take the square root of the result from Step 5.
=√3.5
=1.87
The standard
deviation is approximately 1.87
Example2: (For sample standard deviation)
Here is the data of the recent score performance
of tail-ender batsman:
9,
2, 5, 4, 12, 7, 8, 11, 9, 3, 7, 4, 12, 5, 4, 10, 9, 6, 9, 14
Calculate
the sample standard deviation of
a score of a tail-ender batsman in 20 matches he played in his overall career.
Solution:
Step 1: To Calculate the mean
of the data. Sum all the numbers and divide it by the total number of terms in
the data.
Mean
(Âµ) = (9 + 2 + 5 + 4 + 12 + 7 + 8 + 11 + 9 + 3 + 7 + 4 + 12
+ 5 + 4 + 10 + 9 + 6 + 9 + 14)/20
= 150/20
= 7.5
Step 2: Subtract every number of data from the mean and squaring.
(9 - 7)^{2} = (2)^{2} = 4
(2 - 7)^{2} = (-5)^{2} = 25
(5 - 7)^{2} = (-2)^{2} = 4
(4 - 7)^{2} = (-3)^{2} = 9
(12 - 7)^{2} = (5)^{2} = 25
(7 - 7)^{2} = (0)^{2} = 0
(8 - 7)^{2} = (1)^{2} = 1
(11 - 7)^{2} = (4)^{2} =
16
(9 - 7)^{2} = (2)^{2} = 4
(3 - 7)^{2} = (-4)^{2} =
16
(7 - 7)^{2} = (0)^{2} = 0
(4 - 7)^{2} = (-3)^{2} = 9
(12 - 7)^{2} = (5)^{2} = 25
(5 - 7)^{2} = (-2)^{2} = 4
(4 - 7)^{2} = (-3)^{2} = 9
(10 - 7)^{2} = (3)^{2} = 9
(9 - 7)^{2} = (2)^{2} = 4
(6 - 7)^{2} = (-1)^{2} = 1
(9 - 7)^{2} = (2)^{2} = 4
(14 - 7)^{2} = (7)^{2}=
49
Step 3: To Calculate the mean
of the squared differences.
= (4+25+4+9+25+0+1+16+4+16+0+9+25+4+9+9+4+1+4+49)/19
= 218/19
= 11.47
This value is the sample variance. The sample variance is 11.47
A standard deviation calculator is an alternate way to calculate
standard deviation of the sample and
population data values.
Applications of
Standard Deviation:
Standard
deviation recognizes the variation of a data set in different applications,
including academia,
business, finance, forecasting, manufacturing, medicine, polling, population
traits, etc. It
also enables us how to use different tools in finding the coefficient of
variation, hypothesis testing, and confidence intervals.
1.
Finance and Economics: To gauge risk and volatility, the standard
deviation is widely utilized in finance and economics. As an illustration, the standard
deviation is frequently used in finance to assess the volatility of stock
prices or returns. Greater price volatility is indicated by larger standard
deviation readings, which might suggest increased risk.
2.
Experimental
Research and Data Analysis: In experimental research and data analysis, the
standard deviation is a key factor. The variability or dispersion of data
points around the mean is measured using this technique. Standard deviations
are frequently calculated by researchers to assess the validity of their
findings.
3.
Descriptive
Statistics: In descriptive statistics, the standard deviation is a key
indicator of dispersion. It aids in distilling the distribution of data points
within a dataset. The standard deviation enables determining whether a dataset
has a larger or fewer level of variance when comparing several datasets or
subsets of data.
Summary
The standard deviation is the variation in the measurement of the differences of each value taken from the mean value of data after observing its types formulas, and examples as well. Moreover, if the differences themselves were summed up, the positive will firmly balance the negative and so their sum would be zero.