1. 1.3 Measuring Center & Spread, The Five Number Summary & Boxplots
Describing
Quantitative Data
with Numbers

2. 1.3 I can…
Calculate and interpret measures of center (mean, median) in context.
Calculate and interpret measures of spread (IQR, range, standard deviation) in context.
Identify outliers using the 1.5 IQR rule.
Make a boxplot.
Selecta ppropriate measures of center and spread.
Use appropriate graphs and numerical summaries to compare distributions of quantitative variables.

3. Measuring Spread
Can two dotplots with the same center and same shape, have different spreads? Explain.

4. 5-number Summary & Boxplot
Minimum
Q1 – Lower Quartile
Median
Q3 – Upper Quartile
Maximum

5. Calculations Measures of Center Measures of Spread Outliers Mean
Median
Measures of Spread
Range
IQR
Standard deviation & variance (later)
Outliers

6. Measuring Center Mean…average value…balance point
Median…typical value…midpoint
When are the mean and median close to the same?

7. Resistance
Which is more resistant to extreme values, the mean or median?

8. Center isn’t enough
The correct mean concentration of hair dye isn’t good enough if some boxes are extremely weak and others are extremely strong.
The correct mean weight of a football isn’t good enough if some are extremely light and some are extremely heavy.

9. Skewed Distributions
In a right skewed distribution, where is the mean compared to the median?
In a left skewed distribution, where is the mean compared to the median?

10. Range and Interquartile Range
Range…Hi-Low
Interquartile Range…IQR = Q3 - Q1
See pg. 56
Which is a more resistant measure of spread, range or IQR? Explain.
Be careful about determining the range if outliers are present.

11. Mean or Median? Range or IQR?
The mean is sensitive to a few extreme values while the median is not.
A statistician could have his head in an oven and his feet in ice, and he will say that on average he feels fine.
The range is sensitive to extreme values while the IQR is not.

12. Example: Amt of fat in McDonald’s beef sandwiches
9, 12, 19, 19, 23, 24, 26, 26, 28,
29, 39, 39, 40, 42

13. Boxplots
Make a boxplot for the McDonald’s beef sandwiches data on amount of fat.
Assess the center, spread, symmetry, and skewness from the boxplot.
Boxplots are a way to visualize the 5-number summary.
Boxplots do not show the mode like other graphs.

14. Comparing Distributions
Boxplots show less detail than histograms or stemplots, so they are best used for side-by-side comparison of more than one distribution.

15. Outliers 1.5 x IQR If an observation falls more than
1.5 x IQR above the third quartile or below the first quartile, it can be considered an outlier.
Outliers may reveal interesting information or they may reveal errors.

16. Comparing Quiz Grades Class A 10, 5, 6, 5, 6, 7, 8, 5, 6, 2
Class B 2, 10, 10, 4, 2, 5, 1, 10, 9, 7

17. Comparing Quiz Grades Class A Class B Min 2 1 Q1 5 2 Median 6 6
Max
Outliers?

18. Comparing Quiz Grades Side-by-Side Boxplots
Make the boxplots.
Which class did better?

19. One more measure… Exploring Data- Shape, Outliers, Center, Spread
Graphically
Histogram, pie chart, dotplot, stemplot, back to back stemplot, boxplot
Numerically
Measures of center
mean, mode, median
Measures of Spread or Variability
range, IQR
standard deviation & variance

20. Standard Deviation
Standard deviation measures spread by looking at how far the observations are from the mean.
Be able to interpret the standard deviation…It is roughly the average distance each data value is from the mean of the distribution.

21. Standard Deviation Formula
The standard deviation is the square root of the average squared difference from the mean.

22. Comparing Quiz Grades Class A 10, 5, 6, 5, 6, 7, 8, 5, 6, 2
Class B 2, 10, 10, 4, 2, 5, 1, 10, 9, 7

23. Standard deviation with a calculator
Understanding what the standard deviation means is more important than being able to calculate it by hand.
Use s if data is from a sample, and use ∂if data consists of an entire population.

24. Questions about standard deviation
Standard deviation is a measure of spread about the mean as center, so…
-When is the standard deviation 0?
-What makes the standard deviation larger?
Standard deviation should only be used when the mean is chosen as the appropriate measure of center
S = 0 only when there is no spread. This happens only when all observations have the same value; therefore, s > 0. Why can we not have s < 0?
As our data becomes more spread out about the mean, s increases.
S is not resistant, much like the mean. Outliers have a big impact on standard deviation.

25. What to use and when…
Which is best used to describe
symmetric distributions without outliers (such as the Normal distribution)?
skewed distributions?
Mean with standard deviation
(not resistant)
Median with range & IQR (resistant)
*Numerical summaries do not fully describe the shape of a distribution. Always plot your data!

26. Variance Variance is the standard deviation squared.
Standard deviation is the square root of the variance.

27. Find & interpret the variance for Class A and for Class B

28. Exit Slip
The data below indicates the level of phosphate (in milligrams per deciliter) in a patient’s blood over six trips to the hospital.
Calculate the mean, standard deviation, and variance of the sample size.