Transcript Chapter 1: The Nature of Probability and Statistics

Chapter 3 –
Data Description
Section 3.1 –
Measures of Central Tendency
Measures of Central Tendency
“’Average’ when you stop to think about it is a funny
concept. Although it describes all of us it describes
none of us…While none of us wants to be the average
American, we all want to know about him or her.”
The average American man is 5’9” tall.
The average American woman is 5’3.6”.
The average American is sick in bed 7 days a year
missing five days of work.
On the average day, 24 million people receive animal
bites
By his or her 70th birthday, the average American will
have eaten 14 steers, 1050 chickens, 2.5 lambs and
25.2 hogs.
Measures of Central Tendency
Average does not mean exactly what you think
it means…
An average could indicate several different
things
 Mean
 Median
 Mode
 Midrange
Also, an ‘”average” could be describe a sample
or a population
Measures of Central Tendency
Statistic –

a characteristic or measure obtained by using data
values from a SAMPLE.
 We generally use ROMAN letters to represent
statistics (A, B, C, D …)
Parameter –

a characteristic or measure obtained by using data
values from a POPULATION.
 We generally use GREEK letters to represent
parameters (σ, β, α, µ …)
Measures of Central Tendency
Mean –
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the sum of the values, divided by the total number of
values.
this is usually what you are talking about when you
say average
X 1  X 2  ...  X n 1  X n X
X

n
n
x-bar is used to represent the sample mean.
µ is used to represent the population mean.
Measures of Central Tendency
The Median (MD)
arrange observations from smallest to
largest.
median is either the middle number or the
mean of the middle two numbers.
Measures of Central Tendency
The Mode

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the data value that occurs most frequently
Unimodal- data set has only one value with
the greatest frequency.
Bimodal- data set has 2 values with the
greatest frequency.
Multimodal- data set has more than 2 values
with the greatest frequency.
No mode- no data value occurs more than
once
Measure of Central Tendency
Modal Class- class with the largest
frequency
Pg. 112 example 3-12
Measures of Central Tendency
The Midrange

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the mid point of a data set
MR= (min + max) / 2
Measures of Central Tendency
Rounding Rules –
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In general, round to one place after the last
place given in the data.
ex. 3.45, 5.21, 6.89, 4.22
round to three decimal places.
Measures of Central Tendency
The Median and Mode are resistant,
meaning unusually large or small values
do not affect it.
The Mean and Midrange are not. The one
huge house in the neighborhood allows
the mean home value to skyrocket.
Measures of Central Tendency
GROUPED DATA
When data is grouped in a distribution or
in a graph things are slightly different.
First, make a distribution table with these
column headings

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Class
Frequency
Midpoint
Then add one more column

frequency times midpoint
Measures of Central Tendency
GROUPED DATA
Look at the procedure table on pg. 108
Finding the mean for grouped data
Measures of Central Tendency
GROUPED DATA
Turn to page 107 and look at example 3-3.
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Remember: Midpoint = Upper limit + lower
limit / 2
You will add your f●xm altogether and then
divide by the total frequency (n).
Measures of Central Tendency
GROUPED DATA
Weighted Mean

found by multiplying each value by its
corresponding “weight” and dividing the sum of the
products by the sum of the weights.
For example
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Grade Point Averages in College
A = 4 points, B = 3 points, C = 2 points, D = 1 point.
Each class has a different weight…
Example 3-17 pg. 115
Class
Credits (w)
Grade (x)
English Composition 1
3
A (4 points)
Introduction to Psychology
3
C (2 points)
Biology 1
4
B (3 points)
Physical Education
2
D(1 point)
X = ΣwX
Σw
=
wX
Distribution Shapes (pics pg. 59)
Bell-Shaped- a single peak and tapers off
at either end
Uniform- flat or rectangular
J-Shaped- few data values on left side,
increases from left to right
Reverse J-shaped- few data values on
right side, decreases from left to right
Bimodal- has 2 peaks of the same height
U-shaped- shaped like a U
Distribution Shapes (pg. 117 for
pics)
Positively Skewed or Right-Skewed
 Majority of data values fall on the left of the mean and
cluster at the lower end
 Tail is to the right
Symmetric Distribution
 Data values evenly distributed on both sides of the
mean
Negatively Skewed or Left-Skewed
 Data values fall on right of the mean and cluster at
the upper end.
 Tail is to the left
Practice!
Pg. 118
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2, 3, 7, 10, 12, 13