central tendency spread

Central Tendency Questions


In this demonstration you will be presented with a problem. Attempt to answer each question before you advance to the next screen where you will be given the correct answer along with an explanation.

Demo #1

Demo #1Students take a survey about the effectiveness of their instructor’s teaching. After collecting the data you construct a distribution that looks like the following:


Which would be the most appropriate measure of central tendency and why?


Which would be the most appropriate measure of central tendency and why?A. MeanB. MedianC. Mode

The Median because it is much less affected by the outliers in the tail on the right side of the distribution.

Imagine the distribution had been normal.

Imagine the distribution had been normal.

Mean = 5 Median = 5

The mean and median are the same.

Watch what happens in a positively skewed distribution to the mean.


Median = 5

Mean = 8


Median = 5

Mean = 8

The Mean is more influenced by outliers than is the Median. In fact the Median is considered to be outlier resistant.

Let’s look at the problem again and explain to yourself the reason behind the correct answer:


Students take a survey about the effectiveness of their instructor’s teaching. After collecting the data you construct a distribution that looks like the following:



Which would be the most appropriate measure of central tendency and why?A. MeanB. MedianC. Mode

Demo #2

What if, after a one week weight loss program, you collect data and your weight loss / weight gain distribution looks like the following.


If someone asked how it went for your clients, which central tendency statistic would provide the best description of how it went?

A. MeanB. MedianC. Mode

The Mode, because the mode indicates which is or are the most frequent observation(s). In this case the most frequent observations are 2 pounds lost and 1.5 pounds gained. The Mean or Median of zero does not tell you much about this distribution, neither does standard deviation.

The Mode, because the mode indicates which is or are the most frequent observation(s). In this case the most frequent observations are 2 pounds lost and 1.5 pounds gained. The Mean or Median of zero does not tell you much about this distribution, neither does standard deviation.

Note- generally you do not rely on just one statistic, but several to help tell the story of what is going on.

Here is the problem again. Try to explain in your own words why the Mode is the correct answer.


If someone asked how it went for your clients, which central tendency statistic would provide the best description of how it went?

A. MeanB. MedianC. Mode

Demo #1

A student wants to know the highest and lowest score on a test, to see if his score is closer to one or the other.


Which measure of spread would best answer his question?


Which measure of spread would best answer his question?

A. Inter Quartile RangeB. RangeC. Standard DeviationD. Variance

The Range Remember that the range indicates the lowest and highest score.


Which would be the most appropriate measure of spread and why?


Let’s start from the bottom. Could it be the variance?



Well . . . the variance is the average squared deviations from the mean. Since it uses the mean in its calculations and since the mean moves with the tail (in this case up), then the variance would be an inflated measure of spread.

How about the standard deviation?


The standard deviation is simply the square root of the variance, so it will also be inflated.

How about the range?


The range just tells you the survey with the highest score (most favorable to the teacher) and the lowest score (least favorable to the teacher).

How about the Inter quartile Range?


The interquartile range divides the distribution up into four parts.


Median 50% of the scores above50% of the scores below



25% of scores

25% of scores

25% of scores

25% of scores


And calculates where the middle 50% of the scores are located.


25% of scores

25% of scores

25% of scores

25% of scores

The interquartile range is not affected by outliers

25% of scores

25% of scores

50% of the center scores are within the Interquartile

range

Answer the question again and explain in your own words the reason behind the correct answer:

Answer the question again and explain in your own words the reason behind the correct answer:Students take a survey about the effectiveness of their instructor’s teaching. After collecting the data you construct a distribution that looks like the following:

Which would be the most appropriate measure of spread and why?


Demo # 1

If most of the students do very well on an exam but only a few do very poorly, what would the distribution look like?

A. Negatively SkewedB. NormalC. Positively Skewed

Let’s imagine the 14 students in the class got the following scores:


98 98 97 97 97 97 96 96 96 95 95 94 62 43


98 98 97 97 97 97 96 96 96 95 95 94 62 43

Note that as is stated in the problem in this data set most of the students did very well on the exam and a few did poorly. Let’s create a distribution


98 98 97 97 97 97 96 96 96 95 95 94 62 43


989796956243 94


98 98 97 97 97 97 96 96 96 95 95 94 62 43


What do you think? The direction of the skew is toward the tail.

989796956243 94

Since the tail is on the left or negative side, the curve is negatively skewed.

989796956243 94

Since the tail is on the left or negative side, the curve is negatively skewed.

Here is what the curve (a model of reality that is easier to make sense of) would look like:

Look at the question again and explain why the correct answer is correct in your own words.


If most of the students do very well on an exam but only a few do very poorly, what would the distribution look like?


Demo #2

If most of the students do very poorly on an exam but only a few do very well, what would the distribution look like?



97 78 47 46 46 45 45 45 44 44 44 44 43 43


97 78 47 46 46 45 45 45 44 44 44 44 43 43Let’s create a distribution



977843 44 45 46 47



What do you think?

977843 44 45 46 47



What do you think? The direction of the skew is toward the tail.

977843 44 45 46 47

Since the tail is on the right or positive side, the curve is positively skewed.

977843 44 45 46 47

Since the tail is on the right or positive side, the curve is positively skewed.

Here is what the curve (a model of reality that is easier to make sense of) would look like:


If most of the students do very poorly on an exam but only a few do very well, what would the distribution look like?


Demo #3

A coach of a basketball team is given the mean and the median height of his next opponent. The mean is 6 feet 4 inches but the median height is 6 feet 6 inches. Based on this information answer the following:


What is the nature of the distribution?


What is the nature of the distribution?A. Negatively SkewedB. NormalC. Positively Skewed



Is the team mostly taller or shorter than 6 feet 6 inches tall?



Is the team mostly taller or shorter than 6 feet 6 inches tall?D. ShorterE. Taller

Remember that the mean and median of a normal distribution is the same:


Normal Distribution


Normal Distribution

Mean = 5 Median = 5


Normal Distribution

Scores Evenly Distributed

Mean = 5 Median = 5

However, when the Mean is smaller than the Median, this means that a score to the left of the Median has pulled the Mean to the left or to the negative side, making the distribution negatively skewed.


Negatively Skewed

Distribution


Negatively Skewed

Distribution

Mean = 3 Median = 5


Negatively Skewed

Distribution

Mostly Higher Scores

Mean = 3 Median = 5

When the Mean is calculated it takes into account the distance of all of the scores in its calculation. That makes it sensitive to scores that are far away from the center (outliers).

When the Mean is calculated it takes into account the distance of all of the scores in its calculation. That makes it sensitive to scores that are far away from the center (outliers).

The Median does not take into account the distance of the scores from the center. It just accounts for the rank order of the scores regardless of how far apart they are from one another.

So an outlier like 97 will pull the mean toward it (to the left or the negative direction) but the Median will stay the same whether that score were 97 or 48.

So an outlier like 97 will pull the mean toward it (to the left or the negative direction) but the Median will stay the same whether that score were 97 or 48


97 78 47 46 46 45 45 45 44 44 44 44 43 43Median = 45


97 78 47 46 46 45 45 45 44 44 44 44 43 43Median = 45

48 78 47 46 46 45 45 45 44 44 44 44 43 43Median = 45

One way to tell if the distribution is negatively skewed, normal, or positively skewed is to subtract the Mean from the Median.


• If the result is negative than the distribution is negatively skewed and most of the scores are on the upper end of the distribution.



• If the result is positive than the distribution is positively skewed and most of the scores are on the lower end of the distribution.



Negatively Skewed

Distribution

Mean = 3 Median = 5

-2



Negatively Skewed

Distribution

Normal Distribution

Mean = 3 Median = 5

Mean = 5 Median = 5

-2 0



Negatively Skewed

Distribution

Normal Distribution

Positively Skewed

Distribution

Mean = 3 Median = 5

Mean = 5 Median = 5

Mean = 7 Median = 5

-2 0 2

So a basketball team with the Mean (6’4”) smaller than the Median (6’6”) will have most of the players bunched up by the median with a few outliers below (e.g., 5’6” or 5’8”). This is negatively skewed because the mean is pulled toward the tail.





D. Shorter E. Taller




Is the team mostly taller or shorter than 6 feet 6 inches tall?D. ShorterE. Taller

Kurtosis

Demo #1

What is the nature of the distributions of each of the data sets below?


Data Set A - 1 1 1 2 2 2 2 3 3 6 9A. LeptokurticB. Mesokurtic (Normal)C. Negatively SkewedD. PlatoKurticE. Positively Skewed


Data Set B - 1 2 3 4 5 6 7 8 9 10 11A. LeptokurticB. Mesokurtic (Normal)C. Negatively SkewedD. PlatoKurticE. Positively Skewed


Data Set C - 6 6 6 6 6 7 7 7 7 7 7A. LeptokurticB. Mesokurtic (Normal)C. Negatively SkewedD. PlatoKurticE. Positively Skewed


Data Set D - 3 4 5 5 6 6 6 7 7 8 9A. LeptokurticB. Mesokurtic (Normal)C. Negatively SkewedD. PlatoKurticE. Positively Skewed

Let’s see each distribution graphed –




1 2 3 4 5 6 7 8 9 10 11

1

1

2

2

1 2

3

3

2

6 9



1 2 3 4 5 6 7 8 9 10 11

1 102 11873 4 5 6 9



1 2 3 4 5 6 7 8 9 10 11

6

6

7

7

6 7

6 7

7

6 7



1 2 3 4 5 6 7 8 9 10 11

5

5

6

6

9

6

4 83 7

7

Answer the question again but in your own words:

central tendency spread

Education

skewed distribution

week weight loss program

central tendency statistic

central tendency questions

instructors teaching

correct answer

frequent observations

lowest score