1. Measure of Central
Tendency
A measure indicating the value to be
expected of a typical or middle data point.

2. The Arithmetic Mean
 A central tendency measure representing the
arithmetic average of a set of observations.
 The Population Arithmetic Mean, µ, = Σx
N
 The Sample Arithmetic Mean, x = Σx
n

3. The Arithmetic Mean
 Calculating the Mean from Grouped Data:
x = Σfx
n
 Calculating the Mean of Grouped Data Using
Codes:
x = x0 + w * Σ(u*f)
n

4. The Arithmetic Mean
 Advantages:
 Its concept is familiar to most of people
 Every data set has one & only one mean
 It is useful for comparison
 Disadvantages:
 It is affected by the extreme values in the data set
 It is tedious to calculate for large data
 It cannot be calculated for grouped data with open-
ended classes

5. The Weighted Mean
 Weighted Mean xw = Σ(w *x)
Σw
Where,
xw = symbol for the weighted mean
w = weight assigned to each observation
Σ(w *x) = sum of the weight of each element
times that element
Σw = sum of all the weights

6. Geometric Mean
 For quantities that change over a period of
time, if we need to know an average rate of
change, the arithmetic mean is inappropriate.
Geometric mean offers useful measure in
such a case.
GM = (product of all x values)1/n
Where n is number of x values

7. The Median
 The median is a single value that measures
the central item in the data set. Half the items
lie above the median, half below it. If the data
set contains an odd number of items, the
middle item of the array is the median. For an
even number of items, the median is the
average
 Median = (n + 1)th item in a data array
2
Where n = number of items in the array

8. The Median
 Calculating Median of grouped data =
m = [ (n + 1)/2 – (F + 1)]*w + Lm.
fm
Where,
m = median, n = total number of items,
F = the sum of all the class frequencies up to, but
not including, the median class,
fm = frequency of observations of the median class
w = the class-interval width
Lm = lower limit of the median class interval

9. The Median
 Advantages:
 Extreme values do not affect the median.
 It is easy to understand and can be calculated
from any kind of data, even for grouped data
with open-ended classes, unless the median
falls in an open-ended class.
 Can be calculated for qualitative data.

10. The Median
 Disadvantages:
 Certain statistical procedures that use the
median are more complex than those that use
the mean.
 To find median value, data first need to be
arranged in ascending order. For large set of
data this could be time consuming.

11. The Mode
The mode is that value most often repeated in the
data set. The mode of grouped data,
M o = LM + [ d1 ]*w
(d1 + d2)
LM= lower limit of mode class
d1 = frequency of the modal class minus the
frequency of the class directly below it
d2 = frequency of the modal class minus the
frequency of the class directly above it
w = width of the modal class interval

12. Dispersion
 The spread or variability in a set of data.
 Measures of Dispersion:
• Range
• Interfractile Range: Quartiles, Deciles,
percentiles
• Variance
• Standard Deviation

13. Range
 The range is the difference between the
highest and the lowest values in a frequency
distribution
 The interquartile range measures
approximately how far from the median we
must go on either side before we can include
one-half the values of the data set.
 Interquartile Range = Q3 – Q1.

14. Variance & Standard Deviation of
Population
 Variance is a measure of the average
squared distance between the mean and
each item in the data set.
 σ2 = Σ(x - µ)2 = Σ x2 - µ2
N N
 The standard deviation is the positive square
root of the variance. It is expressed in the
same units as the data.
 σ = √σ2

15. Variance & Standard Deviation of
Population
 For calculating variance of grouped data
 σ2 = Σf*(x - µ)2 = Σf x2 - µ2
N N
Where, f represents the frequency of the
class and x represents the midpoint.

16. Variance & Standard Deviation of
Sample
 S2 = Σ(x - x)2 = Σ x2 – n*x2
n–1 n–1 n–1
Notice the change in formula, instead of N,
n – I has come as divisor. If we divide by n,
the result will have some bias as an
estimator. Using a divisor of n – 1 gives us an
unbiased estimation.

17. Uses of Standard Deviation
 It helps determining, where the values of a
frequency distribution are located in relation
to mean. (Standard Normal Curve)
 It is also useful in describing how far
individual items in a distribution depart from
the mean of the distribution.