Is this a central tendency - spread - symmetry question

The purpose of this presentation is to assist you
in determining if the question you are working
on is a question of -
Central Tendency, Spread, or Distribution Shape?

Central Tendency

Central Tendency
Spread

Central Tendency
Spread
or

Central Tendency
Spread
Distribution Shape

These three question types have to do with the
way the data in a distribution is laid out.

First let’s review what a distribution is:

We will illustrate what a distribution is with a
data set that describes the hours students’ study

Here is the data set:

Student Hours of
Study

Student Hours of
Study
Bart 1

Student Hours of
Study
Bart 1
Basheba 2

Student Hours of
Study
Bart 1
Basheba 2
Bella 2

Student Hours of
Study
Bart 1
Basheba 2
Bella 2
Bob 3

Student Hours of
Study
Bart 1
Basheba 2
Bella 2
Bob 3
Boston 3

Student Hours of
Study
Bart 1
Basheba 2
Bella 2
Bob 3
Boston 3
Bunter 3

Student Hours of
Study
Bart 1
Basheba 2
Bella 2
Bob 3
Boston 3
Bunter 3
Buxby 4

Student Hours of
Study
Bart 1
Basheba 2
Bella 2
Bob 3
Boston 3
Bunter 3
Buxby 4
Bybee 4

Student Hours of
Study
Bart 1
Basheba 2
Bella 2
Bob 3
Boston 3
Bunter 3
Buxby 4
Bybee 4
Bwinda 5

Student Hours of
Study
Bart 1
Basheba 2
Bella 2
Bob 3
Boston 3
Bunter 3
Buxby 4
Bybee 4
Bwinda 5
Data

Student Hours of
Study
Bart 1
Basheba 2
Bella 2
Bob 3
Boston 3
Bunter 3
Buxby 4
Bybee 4
Bwinda 5
Data Set

Student Hours of
Study
Bart 1
Basheba 2
Bella 2
Bob 3
Boston 3
Bunter 3
Buxby 4
Bybee 4
Bwinda 5
From this data set we
will create a
distribution:

Student Hours of
Study
Bart 1
Basheba 2
Bella 2
Bob 3
Boston 3
Bunter 3
Buxby 4
Bybee 4
Bwinda 5
The X Axis, will be the
number of hours of
study

Student Hours of
Study
Bart 1
Basheba 2
Bella 2
Bob 3
Boston 3
Bunter 3
Buxby 4
Bybee 4
Bwinda 5
Hours of Study

Student Hours of
Study
Bart 1
Basheba 2
Bella 2
Bob 3
Boston 3
Bunter 3
Buxby 4
Bybee 4
Bwinda 5
Hours of Study
1

Student Hours of
Study
Bart 1
Basheba 2
Bella 2
Bob 3
Boston 3
Bunter 3
Buxby 4
Bybee 4
Bwinda 5
Hours of Study
1 2

Student Hours of
Study
Bart 1
Basheba 2
Bella 2
Bob 3
Boston 3
Bunter 3
Buxby 4
Bybee 4
Bwinda 5
Hours of Study
1 2 3

Student Hours of
Study
Bart 1
Basheba 2
Bella 2
Bob 3
Boston 3
Bunter 3
Buxby 4
Bybee 4
Bwinda 5
Hours of Study
1 2 3 4

Student Hours of
Study
Bart 1
Basheba 2
Bella 2
Bob 3
Boston 3
Bunter 3
Buxby 4
Bybee 4
Bwinda 5
Hours of Study
1 2 3 4 5

Student Hours of
Study
Bart 1
Basheba 2
Bella 2
Bob 3
Boston 3
Bunter 3
Buxby 4
Bybee 4
Bwinda 5
Hours of Study
1 2 3 4 5
The Y Axis, indicates
the number of times
the same number
occurs

Student Hours of
Study
Bart 1
Basheba 2
Bella 2
Bob 3
Boston 3
Bunter 3
Buxby 4
Bybee 4
Bwinda 5
Hours of Study
1 2 3 4 5
Number of Occurrences

Student Hours of
Study
Bart 1
Basheba 2
Bella 2
Bob 3
Boston 3
Bunter 3
Buxby 4
Bybee 4
Bwinda 5
Hours of Study
1 2 3 4 5
NumberofOccurrences

Student Hours of
Study
Bart 1
Basheba 2
Bella 2
Bob 3
Boston 3
Bunter 3
Buxby 4
Bybee 4
Bwinda 5
Hours of Study
1 2 3 4 5
NumberofOccurrences
1
2
3

Student Hours of
Study
Bart 1
Basheba 2
Bella 2
Bob 3
Boston 3
Bunter 3
Buxby 4
Bybee 4
Bwinda 5
Hours of Study
1 2 3 4 5
NumberofOccurrences
1
2
3
Central Tendency, Spread, or Distribution Shape?Central Tendency, Spread, or Distribution Shape?

Student Hours of
Study
Bart 1
Basheba 2
Bella 2
Bob 3
Boston 3
Bunter 3
Buxby 4
Bybee 4
Bwinda 5
Hours of Study
1 2 3 4 5
NumberofOccurrences
1
2
3
This is a
distribution

Now that we have reviewed the idea of a
distribution, let’s consider Central Tendency.

Central Tendency

What do Central Tendency Questions look like?

Central Tendency
Represents the point or area in the distribution
where scores tend to predominate or cluster.

Central Tendency
1 2 3 4 5 6 7 8 9
1
2
3
4
5

Central Tendency
1 2 3 4 5 6 7 8 9
1
2
3
4
5
Notice how the
scores
predominantly
cluster between the
values 4 and 6.

Central Tendency
1 2 3 4 5 6 7 8 9
1
2
3
4
5
Notice how the
scores
predominantly
cluster between the
values 4 and 6.
The more scores
there are in a
distribution, the
more they tend to
cluster in the center!

Central Tendency
1 2 3 4 5 6 7 8 9
1
2
3
4
5
Notice how the
scores
predominantly
cluster between the
values 4 and 6.
The more scores
there are in a
distribution, the
more they tend to
cluster in the center!
This is called
Central Tendency

Here are some example problems that are
central tendency oriented:

Example 1
A ski rental shop has asked you to determine the
average number of ski boot rentals during the
month of February.

Example 1
average number of ski boot rentals during the
month of February.
If the information request
deals with average or mean,
this would be classified as a
central tendency question.

Example 2
The Northdell County commissioner has asked
you to determine the median household income
of those in rural areas.

Example 2
you to determine the median household income
of those in rural areas.
deals with the median or
middle score, this would be
classified as a central
tendency question.

Example 3
A pastor of a large congregation wants to know
the most frequent number of total attendees at
his Sunday sermons over a period of six months.

Example 3
A pastor of a large congregation wants to know
the most frequent number of total attendees at
his Sunday sermons over a period of six months.
If the information request deals with
the most common or frequent score
or observation, this would be
classified as a central tendency
question.

Questions like these would be classified as
questions of -

Questions like these would be classified as
questions of -
Central Tendency
Spread
Distribution Shape

We will now consider the idea of “Spread” or
“Dispersion” or “Variability”

We will now consider the idea of “Spread” or
“Dispersion” or “Variability”
Central Tendency
Spread
Distribution Shape

An aggregation of observations will tend to have
a dispersion of scores around the central
tendency.

1 2 3 4 5 6 7 8 9
1
2
3
4
5
An aggregation of observations will tend to have
a dispersion of scores around the central
tendency.

When there is no dispersion and all observations
are the same, the aggregation is considered to
represent a constant.

1 2 3 4 5 6 7 8 9
1
2
3
4
5

1 2 3 4 5 6 7 8 9
1
2
3
4
5
5

1 2 3 4 5 6 7 8 9
1
2
3
4
5
5
NO
Dispersion
Spread or
Variation

If there is dispersion (variation) among the
observations then the measure is considered a
variable.

1 2 3 4 5 6 7 8 9
1
2
3
4
5
If there is dispersion (variation) among the
observations then the measure is considered a
variable.

Here are some example problems that are
spread or dispersion oriented:

Example 1
highest and lowest days for ski boot rentals
during the month of February.

Example 1
highest and lowest days for ski boot rentals
during the month of February.
If the information request deals with
the spread between the lowest and
highest values, this would be
classified as a SPREAD question.

Example 2
you to determine how many house hold
incomes lie between the 25th and 50th
percentile.

Example 2
you to determine how many house hold
incomes lie between the 25th and 50th
percentile.
deals with the amount
between two points we are
dealing with a question of
SPREAD.

Example 3
A pastor of a large congregation has found out that his
parishioners attend his sermons on average 2.3
Sundays per month. He wants to know if the
attendance patterns cluster around 2-3 times a month
or 1-4 times a month.

Example 3
A pastor of a large congregation has found out that his
parishioners attend his sermons on average 2.3
Sundays per month. He wants to know if the
attendance patterns cluster around 2-3 times a month
or 1-4 times a month.
The question deals with how spread
out the individuals are from the
mean (2.3): Either way spread out or
clumped mostly together.

We will now consider the idea of the shape of
the distribution

We will now consider the idea of the shape of
the distribution
Central Tendency
Spread
Distribution Shape

We have illustrated the central tendency of a
distribution and the dispersion of scores within
a distribution.
What these two pieces of information do not tell
us is about the shape of the distribution.

Distributions (aggregations of observations) can
be spread evenly around both sides of the
central tendency, like so:
1 2 3 4 5 6 7 8 9
1
2
3
4
5

Such distributions are considered symmetrical.
1 2 3 4 5 6 7 8 9
1
2
3
4
5

When the outlying scores are on the higher end
of the scale the distribution becomes positively
skewed.
1 2 3 4 5 6 7 8 9
1
2
3
4
5
2120
+

When the outlying scores are on the lower end
of the scale the distribution becomes negatively
skewed.
1 2 3 4 5 6 7 8 9
1
2
3
4
5
2120
_

Here is a summary of normal and
skewed distribution shapes
Normal
Distribution
Positively
Skewed
Distribution
Negatively
Skewed
Distribution
Mostly higher
scores
Scores Evenly
Distributed
Mostly lower
scores

Distributions can also be flat where the scores
are evenly spread out across a scale.
Or they can cluster around a particular point.

Example 1
An instructor has been told that her midterm is very
hard. Most of the students get between 60% and 70%
correct and only a few get much higher than that.
60% 70% 80% 90% 100%

Example 1
An instructor has been told that her midterm is very
hard. Most of the students get between 60% and 70%
correct and only a few get much higher than that.
60% 70% 80% 90% 100%
This question deals with the shape
of the distribution

Example 2
Mr. Jones gives a history test where 25% of the
students get Ds, 28% of the students get Cs, 28%
Bs and 25% As. This evenly spread out
distribution looks like this.
25% 28% 28% 25%
Ds Cs Bs As

Example 2
Mr. Jones gives a history test where 25% of the
students get Ds, 28% of the students get Cs, 28%
Bs and 25% As. This evenly spread out
distribution looks like this.
25% 28% 28% 25%
Ds Cs Bs As
On one level we are
dealing with how the
scores are spread out, but
this also affects the shape
of the distribution.

What type of question are you principally
dealing with?
Central Tendency
Spread
Distribution Shape

Is this a central tendency - spread - symmetry question

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