This document discusses standard deviation and related statistical concepts. It defines standard deviation as a measure of variability around the mean and explains how to calculate it from both ungrouped and grouped data. It also defines related terms like variance, standard error of the mean, and confidence limits of the mean. Standard deviation is calculated using the formula that sums the squared deviations from the mean, divided by n-1. Standard error is the standard deviation divided by the square root of the sample size, and confidence limits refer to ranges around the mean within which we can be certain the population mean falls.

1. Standard Deviation
and
Standard Error
Prof. (Dr) Shahla Yasmin
Dept of Zoology
Patna Women’s College

2. Learning objectives
The students will learn about :
• Range and variability
• Standard deviation
• Calculation of standard deviation from
ungrouped and grouped data
• Variance
• Standard error of mean
• Confidence limits of the mean

3. Range and variability
• Variation occurs in the populations, so the samples
(e.g measurement of height, weight, length etc)
collected from the population shows variability.
• Simplest measure of variability in a sample is called
range.
• Range takes into account only the two most extreme
observations of the sample. So it can be used where
measurements are few. Its use is limited.
• e. g. height of girls (n=10) in the ranges from 4ft to
5.5ft

4. Standard deviation
• First introduced by Karl Pearson in 1893
• Standard deviation is a fundamental property of Normal
probability curve, 68.26% of the observations is included by one
standard deviation on either side of the axis of symmetry
(=mean).
• Therefore, standard deviation is a very useful comparative
measure of variation about a mean value of sample.
• If sample includes the entire population, the symbol of standard
deviation is σ (sigma). It is calculated by the formula
σ = √ ∑ (x- μ )2/N
• Where, x = value of observation
μ = population mean
∑ = the sum of
N = number of sampling units in the population

5. Standard deviation
• It is rare to collect sample from the entire population. So
samples are collected from a portion of a population. In this
case, symbol σ is replaced by ‘s’. The formula for calculating s
becomes
• s = √ ∑ (x- x̅)2 /n-1
• Where x̅ = sample mean
• n = number of sampling units in the sample.
• (x- x̅ ) = deviation from the mean

6. Calculation of standard deviation of
ungrouped data
1. Calculate the mean (simple average of the
numbers).
2. For each number: subtract the mean. Square the
result.
3. Add up all of the squared results.
4. Divide this sum by one less than the number of data
points (n - 1).
5. Take the square root of this value to obtain the
sample standard deviation .

7. Calculation of standard deviation
• Following is the wing length measurements (mm)
• 81,79,82,83,80,78,80,87, 82,82
1. Mean x̅ = ∑ x/n =814/10 =81.40 mm
2. (81-81.4)2 = 0.16
(79-81.4)2 = 5.76
(82-81.4)2 = 0.36
(83-81.4)2 = 2.56
(80-81.4)2 = 1.96
(78-81.4)2 = 11.56
(80-81.4)2 = 1.96
(87-81.4)2 = 31.36
(82-81.4)2 = 0.36
(82-81.4)2 = 0.36
3. ∑ (x- x̅)2 = sum of squares of deviations = 56.4

8. Calculation of standard deviation of
ungrouped data
4. Sum of squares/n-1 = ∑ (x- x̅)2 /n-1 =56.4/9 = 6.27
5. Standard deviation s = √6.27 = 2.50 mm

9. Calculation of standard deviation from
grouped data
• Formula for standard deviation of grouped data is
• s = √ ∑ f (x- x̅)2 /n-1
• Wing length measurements:
• Calculation continued on next slide…..

10. Calculation of standard deviation from
grouped data
Class (x)
mm
Frequency f (x- x̅)2 f (x- x̅)2
68 1 36 36
69 2 25 50
70 4 16 64
71 7 9 63
72 11 4 44
73 15 1 15
74 20 0 0
75 15 1 15
76 11 4 44
77 7 9 63
78 4 16 64
79 2 25 50
80 1 36 36
∑ f (x- x̅)2 = 544
n = 100
s = √ ∑ f (x- x̅)2 /n-1
= √544/99
= √5.49 =2.34 mm

11. Variance
• Variance is the square of standard deviation
• Conversely, standard deviation is the square root of
variance
• s = √s2, and
• s2 = ∑ (x- x̅)2 /n-1

12. Standard Error of Mean
• Standard error of the mean (SEM) measures how far a sample
mean deviates from the actual mean of a population
• S.E. = sample standard deviation/√number of sampling units
• S.E. calculated from previous data of wing length=2.34/√100 =
2.34/10 = 0.234

13. Confidence limits of the mean
• The standard error of the mean shows how good is the estimate
that the sample mean is close to population mean.
• Referring to the normal distribution curve, We are 68% confident
that population mean lies within ± 1 S.E. of sample mean.
• We want to be more sure, so 95% or 99% limits are generally
used. These can be obtained by multiplying S.E. (standard error)
by z score (of Normal probability curve)
• 95% of observations fall within ± 1.96 S.E (z= ± 1.96).
• 99% of observations fall within ± 2.58 S.E (z= ± 2.58).
• The intervals ± 1.96 S.E and ± 2.58 S.E are called 95% and 99%
confidence limits respectively.
• 95% confidence limits of wing lengths are 74± (1.96X0.234)
=74.00±0.459