Date post: | 07-Apr-2018 |
Category: |
Documents |
Upload: | sumita-yadav |
View: | 228 times |
Download: | 0 times |
8/4/2019 2 Central Tendency
http://slidepdf.com/reader/full/2-central-tendency 1/36
Properties – describing quantitative
data
Numerical values of an observation around which
most numerical values of other observations in the
data set show a tendency to cluster or group
Extent to which values are dispersed around the
central value called variation.
Extent of departure of numerical values from
symmetrical distribution around the central valuecalled skew ness
8/4/2019 2 Central Tendency
http://slidepdf.com/reader/full/2-central-tendency 2/36
Requisites of a measure of central
tendency
It should be rigidly defined
It should be based on all the observations
Easy to understand and calculate Should have sampling stability
Should not be unduly affected by extreme
observation
8/4/2019 2 Central Tendency
http://slidepdf.com/reader/full/2-central-tendency 3/36
MEASURES OF CENTRAL
TENDENCY
Averages of PositionThe Mode
The Median
Mathematical Averages
The Mean
The Symmetrical Distribution The Positively Skewed Distribution
The Negatively Skewed Distribution
8/4/2019 2 Central Tendency
http://slidepdf.com/reader/full/2-central-tendency 4/36
Mode
A measure of central tendency Value that occurs most often
Not affected by extreme values
Used for either numerical or categorical data There may be no mode or several modes
0 1 2 3 4 5 6 7 8 9 10 11 12 13 14
Mode = 9
0 1 2 3 4 5 6
No Mode
8/4/2019 2 Central Tendency
http://slidepdf.com/reader/full/2-central-tendency 5/36
Mode
Mode – measure of location recognized by the
location of the most frequently occurring
value of a set of data
Sales during 20 days period
53,56,57,58,58,60,61,63,63,64,64,65,65,67,68,71,71,71,71,74 (ascending order data)
8/4/2019 2 Central Tendency
http://slidepdf.com/reader/full/2-central-tendency 6/36
Mode for frequency distribution
Sales Volume (Class
Interval)
No. of Days (Frequency)
53-56 257-60 4
61-64 5
65-68 4
69-72 4
72 and above 1
Frequency distribution of sales per day
8/4/2019 2 Central Tendency
http://slidepdf.com/reader/full/2-central-tendency 7/36
Mode: The Category or Score with the
Largest frequency(or %)
The mode is always a category or score
The mode is not necessarily the category
with the majority(more than 50% of thecases)
The mode is the only measure of central
tendency for nominal variables Some distributions are bimodal
8/4/2019 2 Central Tendency
http://slidepdf.com/reader/full/2-central-tendency 8/36
Mode for grouped data,
M0 = L + f m – f m-1 h
2 f m – f m-1 – f m+1
8/4/2019 2 Central Tendency
http://slidepdf.com/reader/full/2-central-tendency 9/36
THE MEDIAN – measuring
qualitative characters
The median is a measure of centraltendency for variables which are at leastordinal.
The median represents the exact middleof a distribution.
It is the score that divides thedistribution into two equal parts
8/4/2019 2 Central Tendency
http://slidepdf.com/reader/full/2-central-tendency 10/36
Finding the Median in sorted data
“How satisfied are you with your health insurance?
Responses of 7 Individuals
very dissatisfied
very satisfied
somewhat satisfied
very dissatisfied
somewhat dissatisfied
somewhat satisfied
very satisfied
Total(N) 7
8/4/2019 2 Central Tendency
http://slidepdf.com/reader/full/2-central-tendency 11/36
To locate the median
Arrange the responses in order from lowest to highest
(or highest to lowest): Response
very dissatisfied
very dissatisfied
somewhat dissatisfied
somewhat satisfied ( The middle case =Median)
somewhat satisfied
very satisfiedvery satisfied
_________________________________________________
8/4/2019 2 Central Tendency
http://slidepdf.com/reader/full/2-central-tendency 12/36
Summary :Locating the Median with
N=Odd
The median is the response associated with the
middle case.
You find the middle case by :(N + 1) 2
Since N= 7, the middle case is the (7 + 1)
2, or the 4th case
The response associated with the 4th case is
“somewhat satisfied”. Therefore the median is:
Somewhat satisfied.
8/4/2019 2 Central Tendency
http://slidepdf.com/reader/full/2-central-tendency 13/36
To locate the median (N=Even)
Suicide rates of cities
7.44, 10.00, 12.26, 12.61, 13.38, 14.11, 14.30, 14.78
The median is located halfway between
the two middle cases. When the variable
is interval we can average the two middlecases.
Median = 12.61 + 13.38 = 12.99
2
8/4/2019 2 Central Tendency
http://slidepdf.com/reader/full/2-central-tendency 14/36
Median
Robust measure of central tendency Not affected by extreme values
In an ordered array, the median is the
“middle” number If n or N is odd, the median is the middle number
If n or N is even, the median is the average of thetwo middle numbers
0 1 2 3 4 5 6 7 8 9 10 0 1 2 3 4 5 6 7 8 9 10 12 14
Median = 5 Median = 5
8/4/2019 2 Central Tendency
http://slidepdf.com/reader/full/2-central-tendency 15/36
Age of
Automobiles Frequency Cumalative Frequency
0-4 13 13
4-8 29 42
8-12 48 9012-16 22 112
16-20 8 120
120
Median
class
Median for grouped data
Med = L + (n /2) – cf h
f
8/4/2019 2 Central Tendency
http://slidepdf.com/reader/full/2-central-tendency 16/36
Partition Values:
Quartiles, Deciles, and Percentiles
Quartiles – Divide an ordered data set into 4
equal parts - 2nd Quartile - Median
Deciles – Divide an ordered data set into 10
equal parts - 5th Decile - Median
Percentiles – Divide an ordered data set into
100 equal parts - 50th Percentile - Median
8/4/2019 2 Central Tendency
http://slidepdf.com/reader/full/2-central-tendency 17/36
Quartile for a grouped data,
Qi = L + i(n /4) – cf h; i = 1,2,3
f
Decile for a grouped data,
Di = L + i(n /10) –
cf h; i = 1,2…9 f
Percentile for a grouped data,Pi = L + i(n /100) – cf h; i = 1,2…99
f
8/4/2019 2 Central Tendency
http://slidepdf.com/reader/full/2-central-tendency 18/36
_____________________________
Mean. The arithmetic average obtained byadding up all the scores and dividing by the
total number of scores.___________________________________________________________
8/4/2019 2 Central Tendency
http://slidepdf.com/reader/full/2-central-tendency 19/36
Objectives of an Average
Determine one single value that may be
used to describe the character sticks of
entire series. Facilitate comparison at a particular point of
time
Facilitate statistical inference Helps in decision making process
8/4/2019 2 Central Tendency
http://slidepdf.com/reader/full/2-central-tendency 20/36
The Mean_________________________________________________________________Mean. The arithmetic average obtained by adding up all the scores anddividing by the total number of scores.
_________________________________________________________________
Y = raw scores of the variable y__Y = the mean of y
Y = the sum of all the y scores
N = the number of observations
N
Y Y
C i R i I di Ci i Fi di h M
8/4/2019 2 Central Tendency
http://slidepdf.com/reader/full/2-central-tendency 21/36
CITY
Mumbai
Delhi
Kolkatta
Chennai
Banglore
Hyderabad
Baroda
Chandigarh
Meerut
Bhopal
Honolulu
Jaipur
Patna
Kanpur
Ajmer
Crime RATE per 1000
29.3
28.9
32.936.5
25
14.7
58.4
48.8
12.8
21.8
3.4
6.6
40.6
12.9
19.8
Total 392.4
16.2615
4.392
N
Y Y Mean
Crime Rate in Indian Cities:Finding the Mean
8/4/2019 2 Central Tendency
http://slidepdf.com/reader/full/2-central-tendency 22/36
Sample statistic – a numerical value used as
a summary measure using data of thesample for estimation or hypothesis testing
Population parameter - a numerical value
used as a summary measure using data of
the population
8/4/2019 2 Central Tendency
http://slidepdf.com/reader/full/2-central-tendency 23/36
Mean (Arithmetic Mean)
Mean (arithmetic mean) of data values
Sample mean
Population mean
1 1 2
n
i
i n
X X X X
X n n
1 1 2
N
i
i N
X X X X
N N
Sample Size
Population Size
8/4/2019 2 Central Tendency
http://slidepdf.com/reader/full/2-central-tendency 24/36
Arithmetic mean of ungrouped raw
data
Direct method
Indirect method (short cut method)
8/4/2019 2 Central Tendency
http://slidepdf.com/reader/full/2-central-tendency 25/36
Finding the mean in a frequency distribution
When data are arranged in a frequency distribution, we must
give each score its proper weight by multiplying it by its
frequency. We use the following formula to calculate the mean:
__
Y = Σ f Y N
where
__Y = the mean
f Y = a score multiplied by its frequency
Σ f Y = the sum of all the f Y’s N = the total number of cases in the distribution
8/4/2019 2 Central Tendency
http://slidepdf.com/reader/full/2-central-tendency 26/36
Calculating the Mean from a
Frequency Distribution
# of Children(Y)
0
1
2
3
4
5
6
7
Total
Frequency(f)
12
25
733
333
183
26
15
12
1339
Frequency*Y(fY)
0
25
1466
999
732
130
90
84
3526
6.21339
3526
N
fY Y
8/4/2019 2 Central Tendency
http://slidepdf.com/reader/full/2-central-tendency 27/36
Weighted Mean
Used when values are grouped by frequency or
relative importance
28
87
126
45
FrequencyDays to
Complete
28
87
126
45
FrequencyDays to
Complete
Example: Sample of26 Repair Projects
Weighted Mean Daysto Complete:
days 6.31 26
164
28124
8)(27)(86)(125)(4
w
xwX
i
iiW
8/4/2019 2 Central Tendency
http://slidepdf.com/reader/full/2-central-tendency 28/36
Indirect method
The human resource manager at a city
hospital began a study of the overtime hours
of the registered nurses. Fifteen nurses were
selected at random and following overtime
hours were recorded during a month:
13 13 12 15 17 15 5 12 6 7 12 10 9 13 12
5 9 6 10 5 6 9 6 9 12
8/4/2019 2 Central Tendency
http://slidepdf.com/reader/full/2-central-tendency 29/36
The following distribution gives the pattern of overtime work
done by 100 employees of a company. Calculate the average
overtime work done per employee
Overtime
No. of
Employees
Mid
Value d=(m-A)/5 fd
10-15 11 12.5 -2 -22
15-20 20 17.5 -1 -20
20-25 35 22.5 0 0
25-30 20 27.5 1 20
30-35 8 32.5 2 16
35-40 6 37.5 3 18
12
Arithmetic mean of grouped (classified) data
Direct & Step deviation method)
8/4/2019 2 Central Tendency
http://slidepdf.com/reader/full/2-central-tendency 30/36
30
Geometric Mean Geometric Mean of a set of n numbers isdefined as the nth root of the product of
the n numbers and is used to averagepercents, indexes, and relatives.
The formula is: ( X i > 0)
More directly measures the change overmore than one period
Geometric Mean Arithmetic Mean
1 2G
n
n X X X X
8/4/2019 2 Central Tendency
http://slidepdf.com/reader/full/2-central-tendency 31/36
Relationship between Mean,
Median and Mode
M0 = 3Median – 2Mean
OR
Mean – Mode = 3 (Mean – Median)
8/4/2019 2 Central Tendency
http://slidepdf.com/reader/full/2-central-tendency 32/36
The Shape of Distributions
Distributions can be either symmetricalor skewed, depending on whether thereare more frequencies at one end of the
distribution than the other.
?
8/4/2019 2 Central Tendency
http://slidepdf.com/reader/full/2-central-tendency 33/36
Symmetrical
Distributions
A distribution is symmetrical if thefrequencies at the right and left tails of the distribution are identical, so that if it
is divided into two halves, each will be themirror image of the other.
In a unimodal symmetrical distributionthe mean, median, and mode areidentical.
8/4/2019 2 Central Tendency
http://slidepdf.com/reader/full/2-central-tendency 34/36
1.4. Shape of a Distribution
Describes how data is distributed
Symmetric or skewed
Mean = Median = ModeMean < Median < Mode Mode < Median < Mean
Right-SkewedLeft-Skewed Symmetric
(Longer tail extends to left) (Longer tail extends to right)
8/4/2019 2 Central Tendency
http://slidepdf.com/reader/full/2-central-tendency 35/36
Choosing a Measure of Central Tendency
IF variable is Nominal..
– Mode
IF variable is Ordinal...
– Mode or Median(or both)
IF variable is Interval-Ratio and distribution is
Symmetrical…
– Mode, Median or Mean
IF variable is Interval-Ratio and distribution is
Skewed…
– Mode or Median
8/4/2019 2 Central Tendency
http://slidepdf.com/reader/full/2-central-tendency 36/36
Calculate the mean, median and mode for the
following data pertaining to marks in statistics.
There are 80 students in class and the test is of 140 marks.
Marks more than No. of Students
0 80
20 76
40 50
60 28
80 18
100 9
120 3