Measures of location

LECTURE 2

MEASURES OF LOCATION

Saidgozi Saydumarov Sherzodbek Safarov

QM Module Leaders

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Measures of location

Measures of location

• Mean – the arithmetic average value
• Median – the middle value
• Mode – the most frequent value
• Upper quartile
• Lower quartile

Three data structures

• Untabulated (raw data – sequence of numbers or figures)
• Tabulated (ungrouped)
• Tabulated (grouped)

Untabulated data

Untabulated data

Mon	Tue	Wed	Thu	Fri	Sat	Sun
20	15	13	13	27	24	7

Definition: Untabulated data – data given as a sequence of numbers or figures

Example 1: Daily expenditure

To calculate the mean

Add all the numbers up: 20+15+13+13+27+24+7= 119

Divide by the number of observations (elements): 7

The mean is 119/7 = 17

Untabulated data

Untabulated data

Mean: is the arithmetic average of all observations.

•  

Untabulated data

Untabulated data

Example 2: Commute time from home to university in minutes

The mean of this data is

= 17 minutes

Is it representative of the typical commute time?

•  

Mon	Tue	Wed	Thu	Fri
9	12	10	11	43

Untabulated data

Untabulated data

Median is the centermost value of the data when arranged in order.

50% of the ordered data is less than the median while the other 50% is higher than the median.

The median is the value in the group

value

Median in this example is 11

If there is an even number of observations, the median value is not a whole number, for example 3.5th value. In that case, we take the mean of the 3rd and 4th values.

•  

Mon	Tue	Wed	Thu	Fri
9	12	10	11	43

Untabulated data

Untabulated data

Example 3: Survey of stuffed toy preferences by young children

What is the representative or typical toy that children prefer?

We cannot use mean or median here. The data is nominal.

Mode: the most common or most frequent data.

Mode: Stuffed bear

Stuffed bear	Stuffed elephant	Stuffed dinosaur
9	6	5

Untabulated data

Untabulated data

Example 1: Daily expenditure

Median:

8/2 = 4th value.

7, 13, 13, 15, 20, 24, 27

Median: 15

Mode:

most common observation.

Mode: 13

Mon	Tue	Wed	Thu	Fri	Sat	Sun
20	15	13	13	27	24	7

Untabulated data

Untabulated data

Example 1: Lower and upper quartile

Lower quartile: 25%th of the data: value

Upper quartile: 75%th of the data: value

Median: 50% of the data: or value (Middle quartile)

•  

Mon	Tue	Wed	Thu	Fri	Sat	Sun
20	15	13	13	27	24	7

Untabulated data

Untabulated data

Example 2: Commute time from home to university in minutes

Can you find the mode?

It does not exist

Mon	Tue	Wed	Thu	Fri
9	12	10	11	43

Tabulated (ungrouped) data

Tabulated (ungrouped) data

• Definition: Tabulated data – data placed in a frequency table
• Definition: Ungrouped data – single numbers with frequencies
• Example 4: Consider the frequency table below, giving the number of days a particular amount of TV sets sold over a month.
• Calculate mean, median, mode, lower quartile, and upper quartile

No. of TV sets	3	4	5	6	7	8
No. of days	4	6	7	6	5	2

Tabulated (ungrouped) data

Tabulated (ungrouped) data

No. of TV sets (observation)	No. of days (frequency)	Observation X frequency
3	4	12
4	6	24
5	7	35
6	6	36
7	5	35
8	2	16
TOTAL	30	158

Mean:

 

Tabulated (ungrouped) data

Tabulated (ungrouped) data

Median is the value in an ordered dataset

Median = 5

Note: Use cumulative frequency to find 15.5th value

•  

No. of TV sets (observation)	No. of days (frequency)	Cumulative frequency
3	4	4
4	6	10
5	7	17
6	6	23
7	5	28
8	2	30
TOTAL	30

Tabulated (ungrouped) data

Tabulated (ungrouped) data

Mode: the most frequent data

Mode = 5 (data with the highest frequency: 7)

No. of TV sets (observation)	No. of days (frequency)
3	4
4	6
5	7
6	6
7	5
8	2
TOTAL	30

Tabulated (ungrouped) data

Tabulated (ungrouped) data

Lower quartile and upper quartile

Since the data is even, we split the data into

2 groups using the median as the cut off

No. of TV sets (observation)	No. of days (frequency)	Cumulative frequency
3	4	4
4	6	10
5	7	17
6	6	23
7	5	28
8	2	30
TOTAL	30

Tabulated (ungrouped) data

Tabulated (ungrouped) data

We then apply the median on the separated datasets:

Lower quartile

= 4

Upper quartile

= 6

•  

No. of TV sets (observation)	No. of days (frequency)	Cumulative frequency
3	4	4
4	6	10
5	5	15
TOTAL	15

No. of TV sets (observation)	No. of days (frequency)	Cumulative frequency
5	2	2
6	6	8
7	5	13
8	2	15
TOTAL	15

Tabulated grouped data

Tabulated grouped data

• Example 5: The amount spent on food by 50 people in a particular shop is given in the frequency table below:
• Calculate the mean

Expenditure on food	Number of respondents
£0 or more but under £5	2
£5 or more but under £10	6
£10 or more but under £15	8
£15 or more but under £20	14
£20 or more but under £30	12
£30 or more but under £40	6
£40 or more but under £50	2
Total	50

Tabulated grouped data

Tabulated grouped data

Mean

Expenditure on food	Number of respondents (freq.)	Midpoint (x)	Freq X midpoint
£0 or more but under £5	2	2.5	5
£5 or more but under £10	6	7.5	45
£10 or more but under £15	8	12.5	100
£15 or more but under £20	14	17.5	245
£20 or more but under £30	12	25	300
£30 or more but under £40	6	35	210
£40 or more but under £50	2	45	90
Total	50		995

Essential readings

Essential readings

• Jon Curwin…, “Quantitative methods…”, Ch 5
• Glyn Burton…, “Quantitative methods…”, Ch 2.2-2.3
• Richard Thomas, “Quantitative methods…”, Ch 1.5-1.7
• Mik Wisniewski…, “Foundation Quantitative…”, Ch 7
• Clare Morris, “Quantitative Approaches…”, Ch 6
• Louise Swift “Quantitative methods…”, Ch DD2.