1. Business Decision Making
Measures of Location
LSBM

2. Learning Outcomes
By the end of this session, students should be able to:
Summarise Data into Groups.
Calculate of the value of arithmetic Mean for grouped and ungrouped data.
Calculate the value of Median for grouped and ungrouped data.
Calculate the value of Mode for grouped and ungrouped data.
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3. Data Collection Recall types of Data:
Categorical data: Descriptive / Ranked
eg; Yes/No, Personal Opinion, Likert Scale Data, etc.
Quantifiable data: Continuous / Discrete
eg; Heights, Income, Age, Children, Cars owned, etc.
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4. Data Collection
Following is a list of raw data of heights of employees in an organisation:
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5. Data Collection
An Ordered list of data- heights of employees in an Array in ascending Order:
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6. Frequency Distribution - Ungrouped
Listing out the number of times the value (height) repeats.
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7. Frequency Distribution - Grouped
Listing out Grouped Frequency Table.
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8. Cumulative Frequency Distribution
Listing out Grouped Frequency Table (Less than):
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9. Cumulative Frequency Distribution
Listing out Grouped Frequency Table (More than):
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10. Some Examples for you!
In order to assist management negotiations with the trade unions over piecework rates, the management services department of a factory is asked to obtain information on how long it takes for a certain operation to be completed. Consequently, the members of the department measure the time it takes to complete 30 repetitions of the operation, at random occasions during a month. The times are recorded to the nearest tenth of a minute.
Form a frequency distribution of this sample together with a cumulative distribution.
(Hint: Collate data into groups and use tally frequency distribution)
LSBM

11. Some Examples for you!
2. The factory’s daily outputs, in units, of certain article (A) are recorded during the same month as:
49 47
Tally these data into a frequency distribution using the intervals 30-under 35; 35-under 40; and so on. Subsequently for a cumulative frequency distribution column.
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12. Mean, Median & Mode
These are called the Measure of Location; as their values suggests an effective ‘centre’ from a data set, where discrimination of values becomes a must.
These measures of location can also be useful for the purpose of comparison of distributions as shown in the figure below.
However, they have their own advantages and disadvantages.
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13. Mean
Arithmetic Mean of a set of observation is the total sum of the observation divided by the number of observation. This is the most commonly used measure of location, and commonly referred to as ‘Mean’.
Mean = Sum of Observation / No. of Observation
Advantages: Easy to calculate, useful statistic that can be further employed to find other useful statistics, uses all the values.
Limitations: Extreme – outliers can distort the value, discrete data can have meaningless values, cannot be read from a graph.
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14. Median
Median of a set of observations having odd total is the ‘Middle’ value of the observation, but if the total observation is even, the median if the ‘arithmetic’ mean between the two mid-values of the observation.
Median location = n/2 (when total is Odd)
Median item = {n/2 + (n/2 +1)}/2
Advantages: It’s value is not distorted by extreme outliers, graphical illustration is possible.
Limitations: In grouped frequency distribution, it can only be an estimated, limited use for arithmetic calculation & other statistics.
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15. Mode
Mode, within a set of discrete variable observation, is the one that repeats the most, while in continuous variable observation, mode lies in the class of highest frequency. Ascertaining mode could be done by ordering the data, or looking at the highest frequency class.
Advantages: Easy to calculate & not distorted by extreme values.
Limitations: Cannot be used to further calculate other statistic; can have more than one mode in same set of observation.
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16. Calculation of Mean For Ungrouped Data:
Data of Rainfall for 12 months in XYZ city:
Mean = 4.89in
This means that the average rainfall of this city is 4.89 inches.
Formula:
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17. For Frequency Distribution (Grouped - Discrete):
Calculation of Mean
For Frequency Distribution (Grouped - Discrete):
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18. Calculation of Mean For Frequency Distribution:
Total number of Employees, (f) = 100
Total Number of days, (fx) = 247
Therefore, Mean,
Formula:
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19. For Grouped (Continuous) Data:
Calculation of Mean
For Grouped (Continuous) Data:
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20. For Grouped (Continuous) Data:
Calculation of Mean
For Grouped (Continuous) Data:
HMC
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21. For Grouped (Continuous) Data:
Calculation of Mean
For Grouped (Continuous) Data:
HMC
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22. For Grouped (Continuous) Data: - Mean Height of the employees:
Calculation of Mean
For Grouped (Continuous) Data:
- Mean Height of the employees:
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23. Calculation of Median For Ungrouped Data:
Arranging the monthly rainfall in ascending order:
2.1, 2.8, 3.9, 4.2, 4.5, 4.8, 5.2, 5.3, 5.4, 6.5, 6.8, 7.2
As ‘n’ = 12 (even), n/2th observation is the 6th term and th observation is the 7th .
So,
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24. Calculation of Median For Grouped Data:
Since n = 80, the median value should be the mean of 40th and 41st observation.
Formula for Median Calculation:
Where:
L = Lower boundary of median class
F = Cumulative frequency Upto Median class
f = frequency of median class
i = width of the median class
n = total frequency
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25. Calculation of Median Work Out: Total frequency (n) = 80;
Half of n = 40; this term should be on the class corresponding to CF of ‘55’.
So, L = 175 {class 175 – 180}
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26. Calculation of Median Work Out: Total frequency = 80;
Half = 40 – this term would be on the class corresponding to CF of ‘55’.
So, L = 175 {class 175 – 180}
F = ‘35’ { ONLY UPTO median class}
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27. Calculation of Median Work Out: Total frequency = 80;
Half = 40 – this term would be on the class corresponding to CF of ‘55’.
So, L = 175 {class 175 – 180}
F = ‘35’ { ONLY UPTO median class}
f = ‘20’
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28. Calculation of Median Work Out: Total frequency = 80;
Half = 40 – this term would be on the class corresponding to CF of ‘55’.
So, L = 175 {class 175 – 180}
i = 5 {class interval of median class}
F = ‘35’ { ONLY UPTO median class}
f = ‘20’ ; therefore:
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29. Calculation of Median Work Out: So Median Works out to be:
The Median height of the distribution is cm.
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30. Calculation of Mode For Ungrouped Data: For Grouped Data:
The observation that occurs the most will be the mode of the observation.
(Observation could also be bi-modal, or multimodal).
With Frequency distribution, the observation with highest frequency will be the modal observation
For Grouped Data:
The class which has the highest frequency will be the modal class of the distribution.
It can be calculated using following formula:
Where: L = Lower boundary of modal class
fm = frequency of modal class fm+1 = frequency of post-modal class
fm-1 = frequency of pre-modal class i = width of the median class
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31. Calculation of Mode For Grouped Data:
Using the data utilised for Median calculation:
Where: L = Lower boundary of modal class
fm = frequency of modal class
fm-1 = frequency of pre-modal class
fm+1 = frequency of post-modal class
i = width of the median class
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32. Example for you to workout!!
The following set of data represent the distribution of house prices in two counties. By calculating all three measures of location for the two distributions, compare the position in the two counties in 1990s:
Price Range
(£000s)
No. of houses in H&F county sample
No. houses in Hounslow county sample
65 but under 70
70 but under 75
75 but under 80
80 but under 90
90 but under 110
110 but under 140
140 and upwards 2 5 12 20 14 6 1 4 11 19 15
Hint: Consider 200k as the upper limit for last price range
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