lecture 6-measure of central tendency-example

8/10/2019 Lecture 6-Measure of Central Tendency-example

1/36

Compiled by

Mohmad Mohd Derus

1


2/36

Objectives

To calculate measures of central tendency -

mean, median and mode - for grouped and

ungrouped frequency distributions.


3/36

Objectives

Explain why measures of central tendency

differ in grouped and ungrouped frequency

distributions.


4/36

Measures of Central

Tendency Modethe most frequently occurring observation in a

frequency distribution.

Medianin a frequency distribution, the point at which

50% of the cases fall above and 50% of the cases fall

below.

Midrangethe sum of the lowest and highest values in the

data set divided by 2. Rough estimate of the middle.

Meanthe arithmetic average of the observations in a



5/36

Mode Advantages

Simplest summary of a variable that indicates themost frequently occurring category on the nominal

scale of measurement or most frequently occurringobservation on the interval or ratio scale ofmeasurement.

The mode is not sensitive to extreme values in a

frequency distribution An interpretation of the mode is that it provides

the best guess as to the category an observationmay take on a variable


6/36

Mode

Disadvantages

May not be descriptive of the distribution because

the most common category may not occur veryoften

May not be unique; a frequency distribution mayhave more than one mode

May be overly affected by sampling variation Sensitive to how categories are combined


7/36

Case 1 - 8, 12, 15, 9, 6, 8, 10, 9, 8, 6, 7, 8

the mode is 8

Case 2 - 4, 12, 15,2, 5, 3, 10, 9, 1, 6, 7, 8

the is no mode [do not say that the mode is 0]

Case 3 - 15, 18, 18, 18, 22, 33, 40, 40, 40, 60, 76

the mode is 18 and 40 [this data set is said to be

bimodal]


8/36

Class Frequency

5.5 -10.5 1

10.5-15.5 2

15.5-20.5 3

20.5-25.5 8

25.5-30.5 5

30.5-35.5 6

35.5-40.5 1

Modal class

The mode for grouped data is the modal class. The modal class is the class withthe largest frequency.

The modal class for the above is 20.5-25.5, since it has the largest frequency.

Sometimes, the midpoint of the class is used rather than the boundaries, hence

the mode could be also given as 23.


9/36

,

1

m o

1 2

x L C


10/36

Median

When categories of a variable are ordered, a

measure of central tendency should take

that order into account.

The median does so by finding the value of

the variable corresponding to the middle

case. The median is a positional measure.


11/36

Median

Advantages

Relatively easy to obtain

Based on the whole distribution rather than just asmall portion as is the mode

Not affected by extreme values so it is considered

a resistant statistic

Can be computed when distribution is open ended

at the extremes


12/36

Case 1 - (odd sample n ) 8, 12, 15, 9, 6, 8, 10, 9, 8, 7, 9

6, 7, 8, 8, 8, 9, 9, 9, 10,12, 15

median

Case 2 - (even sample n ) 8, 12, 15, 9, 6, 8, 10, 9, 8, 7, 8,6

6, 6, 7, 8, 8, 8, 9, 9, 9, 10,12, 15

Median = 8 + 9 = 17 = 8.5

2 2


13/36

Class Frequency Cumulative frequency

1 1 1

2 2 3

3 6 9

4 5 14

5 4 18

6 3 21

7 3 24

Procedure to locate median: i. locate the point where 12 values would fall below.

ii. Locate the point where 12 values would fall above.

The 12th and 13th values fall in class 4. Hence the median is 4.


14/36

Class

Boundaries

Frequency Cumulative frequency

5.5-10.5 1 1

10.5-15.5 2 3

15.5-20.5 3 6

20.5-25.5 5 11

25.5-30.5 4 15

30.5-35.5 3 18

35.5-40.5 2 20

n=20Procedure to locate median: i. divide n by 2 to find the halfway point.

ii. find the class that contains the 10th value using the

cumulative freq. distribution (this class is called median

class, it contain the median)

Then, substitute the following formula.


15/36

,

m 1

m

m

f f2

x L Cf


16/36

Class

Boundaries

Frequency Cumulative frequency

5.5-10.5 1 1

10.5-15.5 2 3

15.5-20.5 3 6

20.5-25.5 5 11

25.5-30.5 4 15

30.5-35.5 3 18

35.5-40.5 2 20

n=20

Median class

Lm

C [class width]

fm

fm-1


17/36


18/36


19/36

MeanProperties of the mean

The mode and the median can be computed on

metric data but do not take full advantage of

numeric information inherent in the data. Total sum of the deviations around the mean is

zero.

The mean is the balance point of the distribution. Most effective way of summarizing the center of

interval and ratio level data is to average the

values of the variable


20/36

MeanAdvantages

More stable over repeated measures than any other

measure of the center

Other descriptive and inferential statistics arebased on deviations from the mean

Most common statistic

Easily manipulated algebraically Good statistical properties


21/36

MeanDisdvantages

A disadvantage is that the mean is strongly

influenced by extreme measures.

The mean cannot be computed for an open-ended


,


22/36

x f xx or

n n


23/36

The weighted mean is a mean where there issome variation in the relative contribution ofindividual data values to the mean.

Each data value (Xi) has a weight assigned toit (Wi).

Data values with larger weights contributemore to the weighted mean and data values

with smaller weights contribute less to theweighted mean


24/36

Where, Wi = weight Xi = measures


25/36

Each individual data value might actually representa value that is used by multiple people in yoursample. The weight, then, is the number of peopleassociated with that particular value.

Your sample might deliberately over represent orunder represent certain segments of thepopulation. To restore balance, you would place

less weight on the over represented segments ofthe population and greater weight on the underrepresented segments of the population.


26/36

Some values in your data sample might beknown to be more variable (less precise) thanother values. You would place greater weighton those data values known to have greater

precision.


27/36

Supplier

Quantity

Purchased

weightage)

Cost RM)

A 5 bags 12.40

B 8 bags 12.70

C 15 bags 12.80


28/36


29/36

The costs of 3 models of concrete mixers areshown. Find the weighted mean cost of themodels.

Model Number Sold Cost RM)

CM45 10 35000

CM65 5 60000

CM80 8 80000


30/36

The CGPA of 10 QS students who applied fora direct entry to degree programme is shownbelow:

3.15, 3.62, 2.54, 2.81, 3.97, 2.00, 2.76, 2.63, 2.57, 2.33

Find mode, median, midrange, and mean


31/36

For 108 randomly selected contractors, thefollowing IQ frequency distribution was obtained.

Find mean, median and the modal class.

Class Limits Frequency

90-98 6

99-107 22

108-116 43

117-125 28

126-134 9


32/36

To find mean for grouped data:

X=f.Xm

n


33/36

Class

Limits

Class

Boundaries

Frequency

f)

Midpoint

Xm)

f.Xm

n= f.Xm=


34/36

For the following situation, state whichmeasure of central tendency (mean, medianor mode) should be used. The most typical case desired.

The distribution is open-ended. There is an extreme value in the data set.

The data is categorical.

There will be a need to do further statistical computations.

The values are to be divided into two approximately equal

groups, one group containing the larger value and onegroup containing the smaller value.


35/36

Describe which measure of central tendencywas probably used in each situation. Half of the construction workers make more than 5.37

hours per week and half make less than 5.37 hours perweek.

The average number of children per family in the UEM officeis 1.8.

Most worker prefer blue jacket over any other colour.

The average number of times a QS visits the dentist is once

a year. The most common fear today is fear of speaking in public .

The average age of PWD QS in year 2009 is 48.4 years.


36/36

lecture 6-measure of central tendency-example

Documents