Central Tendency 2015

7/25/2019 Central Tendency 2015

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Measures of central

tendencyDr Ismarulyusda bt Ishak


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MEASURES OF CENTRAL TENDENCY FOR

UNGROUPED DATA

Mean

Median

Mode Relationships among the Mean, Median, and Mode

Prem Mann, Introductory Statistics, 7/ECopyright 2010 John Wiley & Sons. All right reserved


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RM 3,500.00


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An indication of

the locationor

centrality of the data.


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The most common measures

Mean

(numerical average),

Median(the midpoint of an order data set such that half of

the data points are above and half are below it)

Mode

(the value that occurs most frequently)


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POPULATION

THE ENTIRE COLLECTION

OF ITEMSTHAT IS THE FOCUS OF

CONCERN.


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POPULATION AND SAMPLE

Popn to identify its characteristics.

Sample to make inferences about the

characteristics of the population


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Figure 1.1 Population and Sample



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Mean

Arithmetic mean (most familiar)

Another name for average.

If describing a population, denoted as , the

greek letter mu.

If describing a sample, denoted as x,called x-bar.

Appropriate for describing measurement data.

Seriously affected by unusual values calledoutliers.


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Calculating Sample Mean

nxx Formula:

That is, add up all of the data points and divide bythe number of data points.

n= sample size

x= variable used to represent individual data

Do not round! Mean need not be a whole

number.


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Population Mean

Mean of all values in population

N

x

N= number of values in population

x= variable used to represent individual data


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Table 3.1 2008 Sales of Six U.S.

Companies


Find the 2008 mean sales for these sixcompanies.


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Example 3-1: Solution

1368228 $228 Billion6

x

x

n

1 2 3 4 5 6x x x x x x x

Thus, the mean 2008 sales of these six companies

was 228, or $228 billion.


149 406 183 107 426 97 1368


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Example 3-3

Table 3.2 lists the total philanthropic givings (in million dollars)

by six companies during 2007.



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Example 3-3

Notice that the charitable contributions made

by Wal-Mart are very large compared to those

of other companies. Hence, it is an outlier.

Show how the inclusion of this outlier affectsthe value of the mean.



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If we do not include the charitable giving's of Wal-Mart (the outlier), the mean of the charitablecontributions of the five companies is

22.4 31.8 19.8 9.0 27.5Mean $22.1 million

5



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Now, to see the impact of the outlier on the

value of the mean, we include the

contributions of Wal-Mart and find the mean

contributions of the six companies. This meanis


22.4 31.8 19.8 9.0 27.5 337.9

Mean $74.73 million6


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Median

Another name for 50th percentile. The middle value when the original data

value are arranged in order of increasing.

Appropriate for describing measurementdata.

Robust to outliers, that is, not affectedmuch by unusual values.


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Example 3-4

The following data give the prices (in

thousands of dollars) of seven houses

selected from all houses sold last month ina city.

312 257 421 289 526 374 497

Find the median.



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First, we rank the given data in increasing orderas follows:

257 289 312 374 421 497 526

Since there are seven homes in this data set andthe middle term is the fourth term,

Thus, the median price of a house is 374.



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Example 3-5

Table 3.3 gives the 2008 profits (rounded to billions ofdollars) of 12 companies selected from all over the world.



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Table 3.3 Profits of 12 Companies for

2008


Find the medianof these data.


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First we rank the given profits as follows:

7 8 9 10 11 12 13 13 14 17 17 45

There are 12 values in this data set. Because

there is an even number of values in the data set,

the median is given by the average of the two

middle values.



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The two middle values are the sixth and seventh

in the foregoing list of data, and these two values

are 12 and 13.

Thus, the median profit of these 12 companies is

$12.5 billion.


12 13 25Median 12.5 $12.5 billion

2 2


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Median

The median gives the centerof a histogram,

with half the data values to the left of the median

and half to the right of the median. The

advantage of using the median as a measure ofcentral tendency is that it is not influenced by

outliers. Consequently, the median is preferred

over the mean as a measure of central tendencyfor data sets that contain outliers or not normal.



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Case Study 3-2 The Gender Pay Gap



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Mode The value that occurs with the highest

frequency in a data set. French word = fashion (most

popular/common)

One data set can have many modes. When two value have same greatest frequency, each

one is a mode and the data set is bimodal

When more than twomultimodal

No value repeatedno mode

Appropriate for all types of data, but mostuseful for categorical data or discrete datawithonly a few number of possible values.


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Speeds (in km/hour) of eight cars that

stopped for speeding violations.

120 140 167 135 140 182 159 133

Solution:

_______ is the mode

l i hi h di


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Relationships among the Mean, Median,

and Mode

1. For a symmetric histogram and

frequency curve with one peak (Figure

3.2), the values of the mean, median, and

mode are identical, and they lie at the

center of the distribution.



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Relationships among the mean, median and

mode

Mean, median, and mode for a symmetric histogram and frequencydistribution curve.

R l i hi h M M di


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and Mode

2. For a histogram and a frequency curveskewed to the right (Figure 3.3), the value ofthe mean is the largest, that of the mode is

the smallest, and the value of the median lies

between these two. (Notice that the modealways occurs at the peak point.) The value ofthe mean is the largest in this case because it

is sensitive to outliers that occur in the righttail. These outliers pull the mean to the right.



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Mean, median, and mode for a histogram and frequency distribution curveskewed to the right.

R l ti hi th M M di


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and Mode

3. If a histogram and a distribution curve

are skewed to the left (Figure 3.4), the

value of the mean is the smallest and thatof the mode is the largest, with the value

of the median lying between these two.

In this case, the outliers in the left tailpull the mean to the left.

Prem Mann, Introductory Statistics, 7/E

Copyright 2010 John Wiley & Sons. All right reserved


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Mean, median, and mode for a histogram and frequency

distribution curve skewed to the left.


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Mean for Grouped Data

N

m f

nx

mf

Mean for

population data

Mean for sample

data

m is midpoint and f is frequency of a class


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Example 3-14

Table 3.8 gives the frequency

distribution of the daily commuting

times (in minutes) from home to workfor all25 employees of a company.

Calculate the mean of the daily

commuting times.

Prem Mann, Introductory Statistics, 7/E

Copyright 2010 John Wiley & Sons. All right reserved


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= 535 = 21.4 minutes

25N

m f

Thus, the employees of this company spend

an average of 21.40 minutes a daycommuting from home to work.


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Choosing Appropriate

If data are symmetric, the mean,

median, and mode will be

approximately the same.

If data are multimodal, report the

mean, median and/or mode for each

subgroup. If data are skewed, report the median.


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Central Tendency 2015