STATISTICS AND NUMERICAL METHODS
MATH 0102
Measures of Central Location
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Mean (Ungroup data)
called the arithmetic mean,
by sharing the sum of the quantities concerned
equally between the numbers of quantities.
RULE: add up the data provided and divide by the
number of quantities.
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Example: What is the arithmetic mean of the
monthly takings?
An investigation into the takings of a small grocer’s shop gives the following
results:
£

January
2 794

February
1 986

March
2 325

April
3 654

May
3 726

June
3 985

July
6 574

August
7 384

September
5 259

October
3 265

November
4 381

December
5 286
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Median (Ungroup data)
Median: that value which divides the data
into two equal halves; 50% of values lying
below and 50% above the median.
Array: place data in numerical order –
whether rising or falling
Median position is
n+1
2
where n = number of values
Median value is that value which corresponds
to the median position
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The median is the value of the middle item of a
distribution once all of the items have been arranged in
order of magnitude.
• The median of the following nine values:
26 4
24
11 12 28 86 90 2
• The median of the following ten values:
• 26 4 24 11 12 28 86 90 2 8
TRY!
1.
2.
Model A: 1, 2, 18, 23, 26, 42, 294
Model B: 43, 44, 45, 69, 73, 76
Find the median?
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Mode (Ungroup data)
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Mode: that value which occurs most often (i.e. with
the highest frequency)
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Example:
a)
b)
c)
7
1 2 1 2 3 4 6 1 2 2 7
1 2 1 2 4 1 1 2 2 7
1 2 3 4 9 5 6 8 7
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Example:
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Example: central location for
ungrouped data

a)
b)
c)
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The following data measures the attention span in
minutes of 15 undergraduates in a sociology
lecture.
4, 6, 7, 8, 8, 8, 8, 9, 9, 10, 11, 12, 14, 15, 18
Find the arithmetic mean
Find the median
Find the mode
Central location: grouped data


Grouped data: data which is only available in
grouped form e.g. class intervals in frequency table
Class mid-points: we assume that the data in any
class interval all fall on the class mid-point. Put
another way, the data are equally spread along any
given class interval
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Mean (grouped data)
j
X
F X
i
i 1
j
F
i 1

i
i
Where Fi = frequency
of ith class interval
Xi = mid-point of
ith class interval
j = number of
class intervals
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Note: simplifying
assumption: all values in
a class interval are
equally spread along that
interval
Find arithmetic mean of grouped data
Daily Demand
Frequency
6-10
8
11-15
4
16-20
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• Find mid-point of each class interval.
Daily
Demand
Frequency
Mid point
FX
6-10
8
8
64
11-15
4
13
52
16-20
6
18
108
18
Arithmetic mean x = ∑ fx / ∑f
= 224/ 18
= 12.4units
224
Median (Ungrouped data)
LCB = lower class boundary
CF = the cumulative number of frequencies in the
classes preceding the class containing the
median
F
= the frequency of the median class
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Mode (Ungrouped data)

Modal class interval: that class interval in which the
mode value falls
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