Analysis of Data

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 Measures of central tendency
● The measures of central tendency locate the
centre of distribution.
● Numerical values in most data sets show a
distinct tendency to cluster around a value of an
observation located somewhere in the middle of
all numerical values.
● It is necessary to know this central value (average)
to describe characteristics of the data set.
● Statistical methods of computing the central value
are called measures of central tendency.
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● However measures of central tendency
do not indicate how items are spread out
on either side of the centre.

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Measures of central tendency

(i) Mean
(ii) Median
(iii) Mode

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(i) Arithmetic mean
● It is also called mean or average.

Mean = ∑X
N

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 Arithmetic mean (A.M.) of ungrouped
data

Q1. Theheights of five runners are 160 cm, 137

cm, 149 cm, 153 cm and 161 cm respectively.
Find the mean height per runner.

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Mean height = Sum of the heights of the
runners/number of runners
= (160 + 137 + 149 + 153 + 161)/5 cm
= 760/5 cm
= 152 cm.
Hence, the mean height is 152 cm.

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Q2. The mean of 40 numbers was found to be
38. Later on, it was detected that a number 56
was misread as 36. Find the correct mean of
given numbers.

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 Solution:
● Calculated mean of 40 numbers = 38.
● Therefore, calculated sum of these numbers
● = (38 × 40) = 1520.
● Correct sum of these numbers
● = [1520 - (wrong item) + (correct item)]
● = (1520 - 36 + 56)
● = 1540.
● Therefore, the correct mean = 1540/40 = 38.5.

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Weighted Arithmetic Mean
● Weighted Mean is an average computed by
giving different weights to some of the
individual values.
● The weights cannot be negative. Some may
be zero, but not all of them; since division by
zero is not allowed.

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Examples of weighted mean
Q3: Suppose that a marketing firm conducts a survey of
1,000 households to determine the average number of
TVs each household owns. The data show a large
number of households with two or three TVs and a
smaller number with one or four. Every household in
the sample has at least one TV and no household has
more than four. calculate the weighted arithmetic
mean.
Number of TVs per Number of
Household Households
1 73
2 378
3 459
4 90
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Step 1: Assign a weight to each value in the dataset:
● x1=1,w1=73
x2=2,w2=378
x3=3,w3=459
x4=4,w4=90
Step 2: Compute the numerator of the weighted mean formula.
● Multiply each sample by its weight and then add the products together:
● Weighted Mean = w1 x1+w2x2+w3x3+w4x4
● = (1)(73)+(2)(378)+(3)(459)+(4)(90)
● =2566
Step 3: Now, compute the denominator of the weighted mean formula by
adding the weights together.
● ∑wi=w1+w2+w3+w4
● = 73 + 378 + 459 + 90
● =1000
Step 4: Divide the numerator by the denominator
● = 2.566
● The mean number of TVs per household in this sample is 2.566 12
(ii) Median
● It may be defined as the middle value (half of the
observations are smaller and half are larger than this
value) in the data set when elements are arranged in
a sequential order of magnitude.

● Thus it is a measure of the location or centrality of the

observations.

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The Median (cont.)

Usually, the median can be found by a simple

counting procedure:
1. With an odd number of scores, list the values in
order, and the median is the middle score in the
list.
2. With an even number of scores, list the values in
order, and the median is half-way between the
middle two scores.

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 Steps in computing the Median of a data set

1. Arrange the data in ascending order.

2. Determine the number of observation n.

3. Determine the observation in the middle of the

data set. If the number of observations is odd, then
the median is the data value that is exactly in the
middle of the data set. That is, it is the observation
that lies in the (n +1)/2 position.

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Example – odd No.

● Find the median of the data set consisting of the

observations 7, 4, 3, 5, 6, 8, 10.
● Solution:

● First, we arrange the data set in ascending order

3 4 5 6 7 8 10.

● Since the number of observations is odd, then

median = (7+ 1)/2 = 4th number in the ordered
list, namely 6. 16
In case of Even no.

● If the number of observations is even, then

the median is the arithmetic mean of the two
middle observations in the data set.
● i.e, it is the arithmetic mean of the data
values that lie in the n/2 and (n/2)+1
position.

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Example….Even no.

● Suppose we have the observations 7, 4, 3, 5, 6, 8,

10, 1. Find the median of this data set.

● First, we arrange the data set in ascending order 1 3

4 5 6 7 8 10. Since the number of the observations n
= 8,

● So by definition the median is the average of the 4th

(n/2 = 8/2 = 4th) and the 5th i.e. Median = (5+6)/2 =
5.5

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Median
 When the number of observations is odd,
Case 1.

the median is the middle value.

Find the median of the following set of points in a game:
15, 14, 10, 8, 12, 8, 16

Solution:
First arrange the point values in an ascending order (or
descending order).
8, 8, 10, 12, 14, 15, 16

Middle position
The number of point values is 7, an odd number. Hence, the
median is the value in the middle position.
Median = 12

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Case 2. When the number of observations is even, the
median is the average of the two middle values.

Example:
Find the median of the following set of points:
15, 14, 10, 8, 12, 8, 16, 13
Solution:
First arrange the point values in an ascending order (or
descending order).
8, 8, 10, 12, 13, 14, 15, 16
The number of point values is 8, an even number. Hence
the median is the average of the 2 middle values.

Sometimes, you may be asked to determine the possible

values of the median of a set of numbers.
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iii) The Mode
● The mode is defined as the most
frequently occurring category or score in
the distribution.
● In a frequency distribution graph, the
mode is the category or score
corresponding to the peak or high point
of the distribution.
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 Mode ….(cont)
● Mode is that value of an observation which occurs with
highest frequency in the raw data or in classified data
set.

● The concept of mode is of great use to large scale

manufacturers of consumable items such as soap,
shampoo, readymade garments etc.

● In all such cases, it is important to know the size that

fits most consumers rather than mean size.
Mode
xample 1: The following is the number of problems that
E

Ms. Matty assigned for homework on 10 different

days. What is the mode?
8, 11, 9, 14, 9, 15, 18, 6, 9, 10

● Solution: Ordering the data from least to greatest, we

get:
6, 8, 9, 9, 9, 10, 11 14, 15, 18

Answer: The mode is 9.

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● Example 2: In a crash test, 11 cars were
tested to determine what impact speed was
required to obtain minimal bumper damage.
Find the mode of the speeds given in miles
per hour below. 24, 15, 18, 20, 18, 22, 24,
26, 18, 26, 24
Solution:
Ordering the data from least to greatest, we
get:
15, 18, 18, 18, 20, 22, 24, 24, 24, 26, 26
Answer: Since both 18 and 24 occur three times,
the modes are 18 and 24 miles per hour. 25
Example 3: A marathon race was completed by 5
participants. What is the mode of these times
given in hours?
2.7 hr, 8.3 hr, 3.5 hr, 5.1 hr, 4.9 hr
Solution: Ordering the data from least to
greatest, we get:
2.7, 3.5, 4.9, 5.1, 8.3
Answer: Since each value occurs only once in the
data set, there is no mode for this set of
data.

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Example 4: On a cold winter day in January, the
temperature for 9 North American cities is
recorded in Fahrenheit. What is the mode of
these temperatures?
-
8, 0, -3, 4, 12, 0, 5, -1, 0
Ordering the data from least to greatest, we
get:
-
8, -3, -1, 0, 0, 0, 4, 5, 12
Answer: The mode of these temperatures is 0.
Let's compare the results of the last two examples. In Example 3, each value occurs only
once, so there is no mode. In Example 4, the mode is 0, since 0 occurs most often in the
set.
Do not confuse a mode of 0 with no mode. 27