A topic of Biostatistics

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