This document discusses measures of central tendency. It defines measures of central tendency as summary statistics that represent the center point of a distribution. The three main measures discussed are the mean, median, and mode. The mean is the sum of all values divided by the total number of values. There are different types of means including the arithmetic mean, weighted mean, and geometric mean. The document provides formulas for calculating each type of mean and discusses their properties and applications.
SAC 25 Final National, Regional & Local Angel Group Investing Insights 2024 0...
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1. CHAPTER
MEASURES OF CENTERAL TENDENCY
Objectives
At the end of this chapter students will be able to:
๏ผ identify types of measure of central tendency
๏ผ define and calculate the mean, mode, median, with their interpretation the result.
๏ผ summarize an aggregate of statistical data by using single measure and make comparison
3.1 Introduction and objectives of measures of central tendency
When we want to make comparison between a groups of number it is good to have a single value that is
considered to be a good representative of each group.
This single value is called the average of the group. Averages are also called measures of central tendency
A measure of central tendency (measures of center or central location) is a summary measure that attempts
to describe a whole set of data with a single value that represents the middle or center of its distribution.
2. Conti.
๏ผSummery Statistic that Represent center point.
๏ผMeasure of central tendency are single value around which a large number of values of distribution
are condensed in the center of the distribution(These measures indicate where most values in a
distribution fall)
3.2.2 objective of central tendency
๏ผTo get single value that describes the characteristics of the entire population
๏ผTo facilitate group compression
๏ผFor further statistical studies
3. Contiโฆ
3.3. General characteristics of measure of central tendency
๏ผ It should be based on all observation
๏ผIt should be as little as affected by extreme observations
๏ผIt should be defined rigidly(have definite value)
3.4. The summation Notation( )
The symbol is used mathematical short hand for operation addition to denote โthe sum ofโ suppose
you have ๐ observation ๐ฅ1,๐ฅ2,โฆ๐ฅ๐ then the sum ๐ฅ1 +๐ฅ2 +,โฆ+๐ฅ๐ can be written as ๐
๐
๐ฅ๐ and read as
the sum of all ๐ฅ๐, ๐ = 1,2,โฆ.,๐
4. Properties of summation
๏ผ ๐
๐
๐ = ๐ โ ๐ , where k is any constant
๏ผ ๐
๐
๐๐๐ = ๐ ๐
๐
๐๐ , where k is any constant
๏ผ ๐
๐
๐ + ๐๐๐ = ๐๐ + ๐ ๐
๐
๐๐ , where a and b are any constant
๏ผ ๐
๐
๐๐ + ๐๐ = ๐
๐
๐๐ + ๐
๐
๐๐
๏ผ ๐
๐
๐๐ โ ๐๐ โ ๐
๐
๐๐ โ ๐
๐
๐๐
Types of Measures of Central Tendency
There are Three different measures of central tendency; each has its advantage and disadvantage.
๏ผ The Mean (Arithmetic, Geometric, Weighted and Harmonic)
๏ผ The Mode
๏ผ The Median
5. Mean
a) Arithmetic Mean
Is defined as the sum of the magnitude of the items divided by the number of items
Is denoted by A.M or ๐
๐ =
๐ฅ1+๐ฅ2+๐ฅ3,โฆโฆ.๐ฅ๐
๐
= ๐=1
๐
๐๐
๐
The mean of ๐, if ๐ฅ1 occurs ๐1 times, ๐ฅ2 occurs ๐2 times, and ๐ฅ๐ occurs ๐๐ times then
The mean will be ๐ฅ = ๐=1
๐
๐๐๐ฅ๐
๐๐
Example: obtain the mean of the following number 2,7,8,2,7,3,7
Solution:
6. Arithmetic mean for grouped data
๏ผ If data are given in the shape of a continuous frequency distribution, then the mean is obtained as
follows
๏ผ The mean will be ๐ฅ = ๐=1
๐
๐๐๐ฅ๐
๐
๐
๐๐
,where ๐ฅ๐ the class mark of the ๐๐กโ class,๐๐ the frequency of the ๐๐กโ
class
7. Special properties of arithmetic mean
1. The sum of deviation a set of items from their mean is always zero. i.e. ๐
๐
๐๐ โ ๐ = 0
2. The sum of the squared deviations of a set of items from their mean is the minimum i.e.
๐
๐
๐๐ โ ๐ 2
< ๐๐ โ ๐ด 2
, ๐ด โ ๐
3. If ๐1 is the mean of ๐1 observation, if ๐2 is the mean of ๐2 observations,โฆif ๐๐ is the mean of
๐๐ observations, then then the mean of all observation in all groups often called the combined mean
is given by:
๐๐ =
๐1๐1+๐2๐2+โฏ๐๐๐๐
๐1+๐2+โฏ๐๐
= ๐
๐
๐๐๐๐
๐
๐
๐๐
8. Conti..
4. if wrong figure has been used when calculating the mean of the correct mean can be obtained
with out repeating the whole processes
Correct mean = wrong mean +
๐๐๐๐๐๐๐ก ๐ฃ๐๐๐ข๐ โ๐ค๐๐๐๐ ๐ฃ๐๐๐ข๐
๐
Where n is total number of observations.
9. Conti.
Example: an average weight of 10 students was calculated to be 65. later on it was discovered that
one weight was misread as 40 instead 80kg. Calculate the correct average weight
5. The effect of transforming original series on the mean.
a) If constant k is added/subtract to or from every observation then the new mean will be the
๐๐๐๐๐ ยฑ ๐ respectively
b) If every observations are multiplied by a constant ๐ then the new mean will be ๐ โ ๐๐๐ ๐๐๐๐
10. Contiโฆ
b) weighted arithmetic mean
๏ผ While calculating simple arithmetic mean, all item were assumed to be equally importance(each value in the data set has
equal weight).
๏ผ When the observation are different weight we use weighted average.
๏ผ Weights are assumed to each item in proportion to items relative importance.
๏ผ If ๐ฅ1, ๐ฅ2,โฆ..,๐ฅ๐ represents value of item, and ๐ค1 ,๐ค2,โฆ., ๐ค๐ are the corresponding weights, then the weighted mean,(๐ฅ๐ค)
given by
(๐ฅ๐ค) =
๐ฅ1๐ค1+๐ฅ2๐ค2+,โฆ..,๐ฅ๐๐ค๐
๐ค1+๐ค2+,โฆ.,๐ค๐
= ๐
๐
๐ฅ๐๐ค๐
๐
๐
๐ค๐
11. Conti..
Merits and demerits of arithmetic mean
Merits:
๏ผ Its based on all observation.
๏ผ It is suitable for further mathematical treatment
๏ผ It is stable average, i.e. Its not affected by fluctuation of sampling
๏ผ It is easy to calculate and simple to understand
12. Contiโฆ
Demerits:
๏ผIts affected by extreme observations
๏ผIts can not be used in the case of open ended classes
๏ผIt can not be used when dealing with qualitative characteristics, such as beauty.
Geometric mean
The geometric mean like arithmetic mean is calculated average. It is used when observed values are
measured as ratios, percentages, proportions, or growth rates.