1. Numerical Methods for Describing Data
Chapter 4
Numerical Methods for Describing Data

2. Population characteristic -
Suppose we want to know the MEAN length of all the fish in Lake Lewisville . . .
Fixed value about a population
Typical unknown
Is this a value that is known?
Can we find it out?
At any given point in time, how many values are there for the mean length of fish in the lake?

3. Statistic - Value calculated from a sample
Suppose we want to know the MEAN length of all the fish in Lake Lewisville.
What can we do to estimate this unknown population characteristic?
Value calculated from a sample

4. Measures of Central Tendency
Mode – the observation that occurs the most often
Can be more than one mode
If all values occur only once – there is no mode
Not used as often as mean & median

5. Measures of Central Tendency
Median - the middle value of the data; it divides the observations in half
To find: list the observations in numerical order
Where n = sample size

6. Suppose we catch a sample of 5 fish from the lake
Suppose we catch a sample of 5 fish from the lake. The lengths of the fish (in inches) are listed below. Find the median length of fish.
The numbers are in order & n is odd – so find the middle observation.
The median length of fish is 5 inches.

7. Suppose we caught a sample of 6 fish from the lake
Suppose we caught a sample of 6 fish from the lake. The median length is …
The median length is 5.5 inches.
The numbers are in order & n is even – so find the middle two observations.
Now, average these two values.
5.5

8. Measures of Central Tendency
Population characteristic
Mean is the arithmetic average.
Use m to represent a population mean
Use x to represent a sample mean
m is the lower case Greek letter mu
statistic
S is the capital Greek letter sigma – it means to sum the values that follow
Formula:

9. Suppose we caught a sample of 6 fish from the lake
Suppose we caught a sample of 6 fish from the lake. Find the mean length of the fish.
To find the mean length of fish - add the observations and divide by n.
Be sure to show students how to use the graphing calculator to insert data into list and to use the calculation function “1-VAR Stats” to calculate the mean and median.

10. YES Now find how each observation deviates from the mean.
The mean is considered the balance point of the distribution because it “balances” the positive and negative deviations.
This is the deviation from the mean. x
(x - x) 3 4 5 6 8 10
Sum
3-6 -3 -2
Find the rest of the deviations from the mean -1
What is the sum of the deviations from the mean? 2
Will this sum always equal zero? 4
YES

11. Imagine a ruler with pennies placed at 3”, 4”, 5”, 6”, 8” and 10”.
To balance the ruler on your finger, you would need to place your finger at the mean of 6.
The mean is the balance point of a distribution

12. 3 4 5 6 8 15 5.5 6.833 The median is . . . The mean is . . .
What happens to the median & mean if the length of 10 inches was 15 inches?
5.5
The median is . . .
6.833
The mean is . . .
What happened?

13. 3 4 5 6 8 20 5.5 7.667 The median is . . . The mean is . . .
What happens to the median & mean if the 15 inches was 20?
5.5
The median is . . .
7.667
The mean is . . .
What happened?

14. NO YES Is the median resistant affected by extreme values?
Some statistics that are not affected by extreme values . . .
Is the median resistant affected by extreme values? NO
Is the mean affected by extreme values?
YES

15. Calculate the mean and median.
Suppose we caught a sample of 20 fish with the following lengths. Create a histogram for the lengths of fish. (Use a class width of 1.)
Mean =
Median =
6.5
6.5
Look at the placement of the mean and median in this symmetrical distribution.
Use scale of 2 on graph
Calculate the mean and median.

16. Calculate the mean and median.
Suppose we caught a sample of 20 fish with the following lengths. Create a histogram for the lengths of fish. (Use a class width 1.)
Mean =
Median =
6.8
5.5
Look at the placement of the mean and median in this skewed distribution.
Use scale of 2 on graph
Calculate the mean and median.

17. Calculate the mean and median.
Suppose we caught a sample of 20 fish with the following lengths. Create a histogram for the lengths of fish. (Use a class width of 1.)
Mean =
Median =
7.75
8.5
Look at the placement of the mean and median in this skewed distribution.
Use scale of 2 on graph
Calculate the mean and median.

18. Recap: In a symmetrical distribution, the mean and median are equal.
In a skewed distribution, the mean is pulled in the direction of the skewness.
In a symmetrical distribution, you should report the mean!
In a skewed distribution, the median should be reported as the measure of center!

19. Trimmed mean: Purpose is to remove outliers from a data set
To calculate a trimmed mean:
Multiply the percent to trim by n
Truncate that many observations from BOTH ends of the distribution (when listed in order)
Calculate the mean with the shortened data set

20. Find the mean of the following set of data.
Find a 10% trimmed.
Mean = 23.8
10%(10) = 1
So remove one observation from each side!

21. What values are used to describe categorical data?
Suppose that each person in a sample of 15 cell phone users is asked if he or she is satisfied with the cell phone service.
Here are the responses:
Y N Y Y Y N N Y Y
N Y Y Y N N
The population proportion is denoted by the letter p.
Pronounced p-hat
What would be the possible responses?
Find the sample proportion of the people who answered “yes”:
60% of the sample was satisfied with their cell phone service.

22. Why is the study of variability important?
Does this can of soda contain exactly 12 ounces?
There is variability in virtually everything
Allows us to distinguish between usual & unusual values
Reporting only a measure of center doesn’t provide a complete picture of the distribution.

23. Notice that these three data sets all have the same mean and median (at 45), but they have very different amounts of variability.

24. Measures of Variability
The simplest numeric measure of variability is range.
Range =
largest observation – smallest observation
The first two data sets have a range of 50 (70-20) but the third data set has a much smaller range of 10.

25. Measures of Variability
What can we do to the deviations so that we could find an average?
Remember the sample of 6 fish that we caught from the lake . . .
They were the following lengths:
3”, 4”, 5”, 6”, 8”, 10”
The mean length was 6 inches. Recall that we calculated the deviations from the mean. What was the sum of these deviations?
Can we find an average deviation?
Another measure of the variability in a data set uses the deviations from the mean (x – x).
Population variance is denoted by s2.
The estimated average of the deviations squared is called the variance.
Degree of freedom

26. Degrees of freedom will be revisited again in Chapter 8.
When calculating sample variance, we use degrees of freedom (n – 1) in the denominator instead of n because this tends to produce better estimates.
Degrees of freedom will be revisited again in Chapter 8.
Suppose that everyone in the class caught a sample of 6 fish from the lake. Would each of our samples contain the same fish?
Would our mean lengths be the same?
See page 189 for more information.
The samples would also have different ranges!

27. Remember the sample of 6 fish that we caught from the lake . . .
Find the variance of the length of fish.
First square the deviations x
(x - x)
(x - x)2 3 -3 4 -2 5 -1 6 8 2 10
Sum
Finding the average of the deviations would always equal 0! 9 4 1 16
What is the sum of the deviations squared?
Divide this by 5. 34
s2 = 6.5

28. Measures of Variability
The square root of variance is called standard deviation.
A typical deviation from the mean is the standard deviation.
s2 = 6.8 inches2 so s = inches
The fish in our sample deviate from the mean of 6 by an average of inches.

29. Calculation of standard deviation of a sample
The most commonly used measures of center and variability are the mean and standard deviation, respectively.
Calculation of standard deviation of a sample
Population standard deviation is denoted by s (where n is used in the denominator).

30. Measures of Variability
Interquartile range (iqr) is the range of the middle half of the data.
Lower quartile (Q1) is the median of the lower half of the data
Upper quartile (Q3) is the median of the upper half of the data
iqr = Q3 – Q1
What advantage does the interquartile range have over the standard deviation?
The iqr is resistant to extreme values

31. Find the interquartile range for this set of data.
The Chronicle of Higher Education ( issue) published the accompanying data on the percentage of the population with a bachelor’s or higher degree in 2007 for each of the 50 states and the District of Columbia.
Find the interquartile range for this set of data.

32. First put the data in order & find the median. 24 26 30
First put the data in order & find the median.
Find the lower quartile (Q1) by finding the median of the lower half.
Find the upper quartile (Q3) by finding the median of the upper half.
iqr = 30 – 24 = 6

33. Another graph- Boxplots
What are some advantages of boxplots?
ease of construction
convenient handling of outliers
construction is not subjective (like histograms)
Used with medium or large size data sets (n > 10)
useful for comparative displays

34. Boxplots When to Use Univariate numerical data
The five-number summary is the minimum value, first quartile, median, third quartile, and maximum value
When to Use Univariate numerical data
How to construct a Skeleton Boxplot
Calculate the five number summary
Draw a horizontal (or vertical) scale
Construct a rectangular box from the lower quartile (Q1) to the upper quartile (Q3)
Draw lines from the lower quartile to the smallest observation and from the upper quartile to the largest observation
To describe
– comment on the center, spread, and shape of the distribution and if there is any unusual features
Use for moderate to large data sets. Don’t use with data sets of n < 10.

35. Draw lines for the whiskers Draw a line for the median
Remember the data on the percentage of the population with a bachelor’s or higher degree in 2007 for each of the 50 states and the District of Columbia.
First draw a scale
Draw a box from Q1 to Q3
Draw lines for the whiskers
Draw a line for the median

36. Modified boxplots
To display outliers:
Identify mild & extreme outliers
An observation is an outliers if it is more than 1.5(iqr) away from the nearest quartile.
An outlier is extreme if it is more than 3(iqr) away from the nearest quartile.
whiskers extend to largest (or smallest) data observation that is not an outlier
Modified boxplots are generally preferred because they provide more information about the data distribution.

37. Draw lines for the whiskers Place a solid dot for the outlier
Remember the data on the percentage of the population with a bachelor’s or higher degree in 2007 for each of the 50 states and the District of Columbia.
To describe:
The distribution of percent of the population with a bachelor’s degree or higher for the U.S. states and District of Columbia is positively skewed with an outlier at 47%. The median percentage is at 26% with a range of 30%.
There is one outlier at the upper end at the distribution, but none at the lower end. Is it extreme?
Draw lines for the whiskers
Place a solid dot for the outlier
First, draw the scale, box and the line for the median
Next calculate the fences for outliers.
24-1.5(6) = 15
30+1.5(6) = 39
30+3(6) = 48

38. Symmetrical boxplots
Approximately symmetrical boxplot
Notice that the range of the lower half and the range of the upper half of this distribution are approximately equal so we can say that it is approximately symmetrical.
Notice that all 3 boxplots are identical, but their corresponding histograms are very different. Can you determine the number of modes from a boxplot?
However, the range of the two halves of this distribution are definitely different sizes, so it would be skewed in the direction of the longest side.
Skewed boxplot

39. Discuss the similarities and differences.
The salaries of NBA players published on the web site hoopshype.com were used to construct the comparative boxplot of salary data for five teams.
Discuss the similarities and differences.
See page 198 for more information.

40. Interpreting Center & Variability
This rule can be used with any distribution – no matter it’s shape!
Chebyshev’s Rule –
The percentage of observations that are within k standard deviations of the mean is at least
where k > 1
If k = 2, then at least 75% of the observations are within 2 standard deviations of the mean.

41. For a sample of families with one preschool child, it was reported that the mean child care time per week was approximately 36 hours with a standard deviation of approximately 12 hours.
Using Chebyshev’s rule, at least 75% of the sample observations must be between 12 and 60 hours (within 2 standard deviations of the mean).
At most, what percent of the observations are greater than hours?
At least 89% of the observations are between 0 & 72 hours. Since time can’t be negative, at most 11% of the observations are above 72 hours.

42. What’s my area?
Input the following command into a graphing calculator in order to graph a normal curve with a mean of 20 and standard deviation of 3.
Y1 = normalpdf(X,20,3) (Window x: [10,30] y: [0,0.2])
Use the command 2nd trace, 7 to find the area under the curve for the: (Round to 3 decimal places.)
Lower limit: 17 Upper limit: 23 Area: ________
Lower limit: 14 Upper limit: 26 Area: ________
Lower limit: 11 Upper limit: 29 Area: ________

43. What pattern do you notice?
What’s my area?
Graph a normal curve with a mean of 50 and standard deviation of 5.
Y1 = normalpdf(X,50,5) (x: [30,70] y: [0,0.1])
Find the area under the curve for the following:
Lower limit: 45 Upper limit: 55 Area: ________
Lower limit: 40 Upper limit: 60 Area: ________
Lower limit: 35 Upper limit: 65 Area: ________
What pattern do you notice?

44. Interpreting Center & Variability
Empirical Rule-
Approximately 68% of the observations are within 1 standard deviation of the mean
Approximately 95% of the observations are within 2 standard deviation of the mean
Approximately 99.7% of the observations are within 3 standard deviation of the mean
99.7%
95%
68%
Can ONLY be used with distributions that are mound shaped!

45. The height of male students at PWSH is approximately normally distributed with a mean of 71 inches and standard deviation of 2.5 inches.
What percent of the male students are shorter than inches?
b) Taller than 73.5 inches?
c) Between 66 & 73.5 inches?
About 2.5%
About 16%
About 81.5%

46. Measures of Relative Standing
Z-score
A z-score tells us how many standard deviations the value is from the mean.
One example of standardized score.

47. What do these z-scores mean?
-2.3
1.8
-4.3
2.3 standard deviations below the mean
1.8 standard deviations above the mean
4.3 standard deviations below the mean

48. Sally is taking two different math achievement tests with different means and standard deviations. The mean score on test A was 56 with a standard deviation of 3.5, while the mean score on test B was 65 with a standard deviation of Sally scored a 62 on test A and a 69 on test B. On which test did Sally score the best?
Z-score on test A
Z-score on test B
She did better on test A.

49. Measures of Relative Standing
Percentiles
A percentile is a value in the data set where r percent of the observations fall AT or BELOW that value

50. In addition to weight and length, head circumference is another measure of health in newborn babies. The National Center for Health Statistics reports the following summary values for head circumference (in cm) at birth for boys.
Head circumference (cm)
32.2
33.2
34.5
35.8
37.0
38.2
38.6
Percentile 5 10 25 50 75 90 95
What percent of newborn boys had head circumferences greater than 37.0 cm?
25%
10% of newborn babies have head circumferences bigger than what value?
38.2 cm