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



Bart 1



Bart 1Basheba 2



Bart 1Basheba 2Bella 2



Bart 1Basheba 2Bella 2Bob 3



Bart 1Basheba 2Bella 2Bob 3Boston 3



Bart 1Basheba 2Bella 2Bob 3Boston 3Bunter 3



Bart 1Basheba 2Bella 2Bob 3Boston 3Bunter 3Buxby 4



Bart 1Basheba 2Bella 2Bob 3Boston 3Bunter 3Buxby 4Bybee 4



Bart 1Basheba 2Bella 2Bob 3Boston 3Bunter 3Buxby 4Bybee 4Bwinda 5




Data




Data Set




From this data set we will create a distribution:







The X Axis, will be the number of hours of

study




Hours of Study




Hours of Study1




Hours of Study1 2




Hours of Study1 2 3




Hours of Study1 2 3 4




Hours of Study1 2 3 4 5









The Y Axis, indicates the number of times

the same number occurs



































Number of Occurrences





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Central Tendency, Spread, or Distribution Shape?Central Tendency, Spread, or Distribution Shape?




Num

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Central Tendency, Spread, or Distribution Shape?Central Tendency, Spread, or Distribution Shape?




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This is a distribution


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


Central TendencyCentral Tendency, Spread, or Distribution Shape?

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


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



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



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

A ski rental shop has asked you to determine the 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

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

If the information request 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 -



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



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



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



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



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


NODispersion 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



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



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



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


Example 1

A ski rental shop has asked you to determine the highest and lowest days for ski boot rentals during the month of February.


Example 1

A ski rental shop has asked you to determine the 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

The Northdell County commissioner has asked you to determine how many house hold incomes lie between the 25th and 50th percentile.


Example 2

The Northdell County commissioner has asked you to determine how many house hold incomes lie between the 25th and 50th percentile.

If the information request 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.



Central Tendency

Spread

Distribution Shape


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.

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


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


Education

Is this a central tendency - spread - symmetry question