1. Describing Quantitative Data with Numbers

2. Section 1.3

3. Mean, median, and mode
What is the difference?

4. The Mean
Note: the x-bar notation only applies to the mean of a sample, not the mean of a population
However, the calculations are the same

5. Let’s Try it I randomly select 4 AP Stats students
For that sample of four people, the individuals have the following GPAs: 3.595, 4.095, 3.214, and 3.524
What is the mean of these data?

6. Let’s Try it I randomly select 4 AP Stats students
For that sample of four people, the individuals have the following GPAs: 3.595, 4.095, 3.214, and 3.524
What is the mean of these data?
3.607
Now let’s remove the 4.095
What happens to the mean?

7. Let’s Try it I randomly select 4 AP Stats students
For that sample of four people, the individuals have the following GPAs: 3.595, 4.095, 3.214, and 3.524
What is the mean of these data?
3.607
Now let’s remove the 4.095
What happens to the mean?
Now 3.444—a fairly large change (change of .163)
What does this tell us about the mean as a way to measure the center of the data?

8. An Alternative: The Median

9. Let’s Try it Same data: 3.595, 4.095, 3.214, and 3.524
What is the median?

10. Let’s Try it Same data: 3.595, 4.095, 3.214, and 3.524
What is the median?
3.5595
Now remove the observation again

11. Let’s Try it Same data: 3.595, 4.095, 3.214, and 3.524
What is the median?
3.5595
Now remove the observation again
Median now is (change of .0355)
What does this tell us about the median as a measure of center?

12. Mean or Median? It depends…
When describing a distribution, median is often more useful
For some calculations, the mean MIGHT be more appropriate
Taxes
Income
Measures that are per capita

13. What about the mode? The least often used—except on standardized tests
Simply the most common value for a variable
So…in our 4-observation dataset of GPA, the mode is not very exciting, because they are all different values
Technically we would have 4 modes
But if we take the age (instead of GPA) of those 4 students, they are (not in order): 16, 17, 17, 17
What is the mode?

14. Beyond the Center
In practice, we often care about much more than just the center of the data
The average temperature is the same in San Francisco as in Springfield (MO)
Despite very different temperatures
What does the mean/median fail to capture?

15. Beyond the Center
In practice, we often care about much more than just the center of the data
The average temperature is the same in San Francisco as in Springfield (MO)
Despite very different temperatures
What does the mean/median fail to capture?
Variability
Can be measured in terms of the range
Any problems with using the range to describe variability?

16. Variability The Range Interquartile range (IQR)
Weakness: depends on the minimum and maximum values
Particularly if they are outliers, this could be a problem
Interquartile range (IQR)
Looks at the range of the middle half (50%) of the data
1st quartile is the point that separates the bottom quarter of data from the second-from-the-bottom
2nd quartile is the median
3rd quartile is the point that separates the top quarter of data from the second- from-the-top

17. Variability

18. Back to the Tennis Serves
124.5, 122.1, 120.3, 119.7, 118.7, 116.5, 115.6, 114.5, 114, 113.9, 113.7, 112.6, 112.4, 112.3, 112.2, 110.5, , 108.3, 107.3, 103.1, 101.9
Find the mean
Find the median
Find the 1st quartile
Find the 3rd quartile

19. Back to the Tennis Serves
124.5, 122.1, 120.3, 119.7, 118.7, 116.5, 115.6, 114.5, 114, 113.9, 113.7, 112.6, 112.4, 112.3, 112.2, 110.5, , 108.3, 107.3, 103.1, 101.9
Find the mean
Find the median
Find the 1st quartile
Find the 3rd quartile 117.6
So the IQR is =

20. Defining Outliers

21. Were there any outliers?
So our IQR was 7.65
1.5*7.65=
On the high end, an observation would have to be ABOVE the 3rd quartile (117.6) =
On the low end, an observation would have to be BELOW the 1st quartile (109.95) =
Were there any outliers?

22. 5-number summary So, let’s do it using the tennis serves:
Min Q1 Med Q Max

23. Boxplots
AP Statistics Height
Min Q1 Med Q3 Max

24. Boxplots
In our example, the median is exactly in between the 1st and third quartiles. This does not always happen
Similarly, you’ll notice that one whisker is longer than the other
This is totally normal
What does that tell us about the skewness of our data?

25. Standard Deviation
Most common way of measuring the spread of a distribution
Essentially measuring how far, on average, the values in the distribution are from the mean
So the mean is important here
If you have reason to think the mean is not ideal, standard deviation might not be ideal either

26. Standard Deviation

27. Standard Deviation
On the AP test, you will be given the formula for standard deviation
You do not need to memorize it
But you do need to understand what the formula means

28. Let’s Try it
Back to our GPA example: find the standard deviation of the following GPAs: 3.595, 4.095, 3.214, and 3.524
Remember, we calculated the mean as 3.607

29. Let’s Try it
Back to our GPA example: find the standard deviation of the following GPAs: 3.595, 4.095, 3.214, and 3.524
Remember, we calculated the mean as 3.607
Standard deviation= .365