OPMT 5515
Quantitative Methods
Introduction & Lecture 1
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About Me
Education/Work
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What is Statistics?
Uncertainty
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Data
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Inference
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How Statistics?
Business
Problem
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Statistics
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Business
Solutions
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Lecture 1: Descriptive Statistics – Measures of Centre
(Mean, Median, Mode, Weighted Mean)
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Population vs. Sample
Population: ALL the elements from a particular
group being examined
• The number of units (data) in a population is
referred to with the variable “N”
• Example: All the BCIT students regardless of
program.
Sample: a SUBSET of the population; one or more
observations drawn from the population
• The number of units (data) in a sample is
referred to with the variable “n”
• Example: HRMG students.
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Central Tendency
The central tendency is a central, or “typical”, value
in the data
The most common metrics for the central tendency
are:
Mean: the average
Median: the middle value
Mode: the most frequent value
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MEAN
ഥ (read “x-bar”) or 𝝁 (read “mu”): the
Mean, 𝒙
average; determined by adding all the data values
together and dividing by the total number of data
Total of all the values
Mean =
Number of values
σ 𝑥𝑖
𝜇=
𝑁
(Population)
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σ 𝑥𝑖
𝑥=
𝑛
(Sample)
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Ex 1
You randomly select 3 students and ask them what their grade was in
Business Math. Jane got 99, Jeff 91, Barney 80.
a) Calculate the average grade for this sample.
x1 = 99 (Jane)
Mean, 𝑥ҧ =
x2 = 91 (Jeff)
x3 = 80 (Barney)
σ 𝑥𝑖
𝑛
99+91+80
270
𝑥ҧ =
=
=
90
3
3
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Ex 1 (cont’d)
b) Fred whispers to you that he only got 6. Recalculate the average,
including Fred’s result
x1 = 99 (Jane)
x4 = 6 (Fred)
x2 = 91 (Jeff)
Updated Mean, 𝑥ҧ =
x3 = 80 (Barney)
σ 𝑥𝑖
𝑛
99+91+80+6
276
𝑥ҧ =
=
=69
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The mean is greatly affected by extremely large or
small values called OUTLIERS.
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Median
෥ (read “x-tilde”): when the data is
Median, 𝒙
ordered from smallest to largest, the median is the
middle value (if there are two middle values, we take
the average of the two values)
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Ex 2
A small apartment building has only 6 households. Their annual
incomes are:
$55,000 $6,700,000 $71,000 $62,000 $60,000 $66,000
a) Calculate the both the mean and median income.
55𝐾 + 6,700𝐾 + 71𝐾 + 62𝐾 + 60𝐾 + 66𝐾 7,014,000
=
6
6
𝑥ҧ = $𝟏, 𝟏𝟔𝟗, 𝟎𝟎𝟎
𝑴𝒆𝒂𝒏: 𝑥ҧ =
Median: First order from smallest to largest
$55,000 $60,000 $62,000 $66,000 $71,000 $6,700,000
62,000 + 66,000
𝑀𝑒𝑑𝑖𝑎𝑛 =
= $𝟔𝟒, 𝟎𝟎𝟎
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Which is more accurate?
If there are extremely large or small values the median is a better measure of central
location
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Mode
Mode, 𝑴: the data value that appears most often
within the set of data
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Ex 3 Listed below are the ages of groups of students.
Determine the mode for each group.
Mode = 20
a) 18 20 20 20 21 25 25
b) 18 20 20 20 21 25 25 25 Mode = 20 and 25
c) 18 18 18 20 20 20 21 21 21 25 25 25
No mode since all numbers occur the same number of times.
If there are no mode, or many modes, it is not a very reliable
measure of the centre
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Weighted Average
ഥ𝒘 : similar to an ordinary arithmetic mean
Weighted Mean, 𝒙
(see above), except that instead of each of the data points
contributing equally to the final average, some data points
contribute more than others. Thus, each data point has a
weighted multiplier, W, applied to it
σ 𝒙𝒘
ഥ𝒘 =
𝒙
σ𝒘
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