1. What is Standard Deviation?
The
quantitative degree by which every value in a given data varies from a Measure
of Central Tendency is known as Standard Deviation (S.D.) It is denoted by
Greek small letter Ïƒ. It is the positive square root of the Arithmetic mean of
squares of deviations of a given data set from their Arithmetic mean.
The termÂ â€˜Standard Deviationâ€™Â was first used byÂ Karl PearsonÂ in 1894.
Question:
What is Measure of Central Tendency?
Answer:
A single value which defines the
properties of a given distribution is known as Measure of Central Tendency.
Mean, mode and median are three main Measure of Central Tendency. Â Mean
is the most frequently used Measure of Central Tendency.
Why is mean used as a Measure of
Central Tendency?
Answer:
Mean is the sum of all the values of
an observation divided by the total number of observations in a given data.
Formula of Arithmetic
MeanÂ
For the Frequency Distribution
Arithmetic Mean is used as a Measure of Central Tendency quite frequently
because of the following merits:-
- It covers all the values of an observation.
- It allows further mathematical treatment.
- The error is minimized by using Arithmetic Mean for
every value.
- Mean is the only value for which the sum of deviations
for an observation is zero.
In case of Standard Deviation, Mean
is represented by x-bar and in case of population observations mean is denoted
as Greek letter mu (Âµ). Unlike Standard Deviation, the formula of mean
remains the same in both the cases.
2. Formula for Standard Deviation
Population Standard
Deviation
Sample Standard
Deviation
Question:
Why are there two different
methods of Standard Deviation S.D.?
Answer:
S
is an unbiased estimator for Population Standard Deviation. However, Sigma
(Ïƒ) is a biased estimator for Population S.D. For large samples, the bias
starts tending to zero as n increases. Hence, sigma is also considered as an
equally important S.D.
For
understanding the concept of Standard Deviation, the understanding of Variance
is important
Question:
What is Variance?
Answer:
Given
a data set, the values vary about a measure of central tendency (mean, median,
mode etc) and these measures are known as measures of variation or dispersion.
Variance is one of the frequently used measures of dispersion where,
For Discrete Data:
Finding Standard Deviation through
Variance:
For finding Standard Deviation in case of ungrouped data (Discrete Data) or grouped data (Continuous Data) one can use, methods (1) and (2) given above respectively.
S.D.
= âˆšVariance |
3. How to Find Out Standard Deviation
In case of Population Values,
procedural steps to find out Standard Deviation are as follows:
Step 1:Â Find Âµ.
Step 2: For each value, subtract the mean from the value and square
it.
Step 3:Â Add all the values obtained through step 1.
Step 4:Â Divide the values obtained through 3 by total number
of observations. (N in case of grouped data and âˆ‘ frequencies in case of
ungrouped data). We obtain the variance after this.
4. Example of Standard Deviation
Question:
For Discrete Data, consider marks
obtained in a class of 10 students (out of 50): 10, 20, 45, 45, 24, 8, 19, 45,
23, 35.
Solution:
Here,
N = 10
The mean of the given Data set is (10 + 20 + 45 + 45 + 24 + 8 + 19 + 45 + 23 +
35)/10 = 27.4
X_{i}
(Observations) |
(x_{i
}- Âµ) |
(x_{i
}- Âµ)^{2} |
10 |
-17.4 |
302.76 |
20 |
-7.4 |
54.76 |
45 |
17.6 |
309.76 |
45 |
17.6 |
309.76 |
24 |
-3.4 |
11.56 |
8 |
-19.4 |
376.36 |
19 |
-8.4 |
70.56 |
45 |
17.6 |
309.76 |
23 |
-4.4 |
19.36 |
35 |
7.6 |
57.76 |
1.N
= 10 |
2.âˆ‘(x_{i
}- Âµ) = 0 |
3.âˆ‘(x_{i
}- Âµ)^{2 }= 1822.4 |
Here, N=10, variance= (3/N) = 182.4
S.D. = âˆšvariance â†’ Standard
Deviation = 13.49963
Here, S.D. is very high which
indicates that there is a very high variation in marks of students in the
class.
Question:
Given no. of people belonging to different age groups. Find the S.D. for grouped data?