1. Class Session #2 Numerically Summarizing Data
Measures of Central Tendency
Measures of Dispersion
Measures of Central Tendency and Dispersion from Grouped Data
Measures of Position

2. Recall the Definitions
Parameter – a descriptive measure of a population
(p = parameter = population,
usually in Greek letters)
Statistic – a descriptive measure of a sample
(s = statistic = sample,
usually in Roman letters)

3. Common “descriptions”
? Average ? – “typical” as described in the news reports
Give some of today’s examples
Data distributions’ “characteristics”
Shape – look at a picture (histogram)
Center – mean, mode, median
Spread – range, variance, std. dev.

4. Central Tendency Definitions
Arithmetic mean – the sum of all the values of the variable in the data set, divided by the number of observations
Population arithmetic mean - computed using all the individuals in the population (“mew” = μ) (≠ micro µ)
Sample arithmetic mean – computed using the sample data (“x-bar”)
Note:  is a statistic, μ is a parameter

5. More Central Tendency Defs
Median – the value that lies in the middle of the data, when arranged in ascending order (think of the median strip of highway in the middle of the road)
Mode – the most frequent observation of the variable in the data set (think “a la mode” in fashion /on top)

6. Measures of Dispersion Definitions
Range (R) – the difference between the largest data value (maximum) & the smallest data value (minimum)
Deviation about the mean – how “spread out” the data is.
? for both population and sample variance, the sum of all deviations about the mean equals what ?
? the square of a non-zero number is ?

7. More Measures of Dispersion Definitions
Population Variance – sum of squared deviations about the population mean, divided by the number of observations in the population N (sigma squared)
? i.e. population variance is the mean of the ______ _________ ____ __ _________ ___ ?
Answer: Population variance is the mean of the squared deviations about the population mean

8. More Measures of Dispersion Definitions
Sample Variance – sum of the squared deviations about the sample mean, divided by the number of observations minus one (s squared)
Degrees of freedom is the “n-1”

9. More Measures of Dispersion Definitions
Population Standard Deviation – the square root of the population variance (sigma, written as “σ”)
Sample Standard Deviation – the square root of the sample variance (s, written as “s”)
BTW, later we discover “s” itself is a random variable

10. Empirical Rule for Symmetric Data
If the distribution is bell shaped:
68% of data within 1 std deviations
95% of data within 2 std deviations
99.7% of data within 3 standard deviations of the mean
Rule holds for both samples & populations

11. Supposing Grouped Data
Approximate mean of a variable from a frequency distribution
Use the midpoint of each class
Use the frequency of each class
Use the number of classes
Population Mean
Sample Mean

12. Supposing Grouped Data
Weighted Mean
Good to use when certain data values have higher importance (or weight)
[Sum of each value of variable times its weight] / [sum of weights]
Examples of Grade Point Average (GPA) and mixed nuts pricing

13. Supposing Grouped Data
Population Variance
sum of [(midpoint – mean)2 times frequency] / [sum of frequencies]
Sample Variance
as before except “-1” in denominator (the degrees of freedom thing again)

14. Supposing Grouped Data
Population Standard Deviation
take square root of population variance
Sample Standard Deviation
take square root of sample variance

15. Measures of Position Definition
z-Score – the distance that a data value is from the mean in terms of standard deviations. Equals (data value minus mean) divided by standard deviation]
Population z-score
Sample z-score

16. Measures of Position Definitions
z-score equals [(data value minus mean) divided by standard deviation]
Is a "unitless" measure
Can be “normalized” to get
Mean of zero
Standard Deviation of one

17. Measures of Position Definitions
z-score purpose is to provide a way to "compare apples and oranges"
by converting variables with different centers and/or spreads
to variables with the same center (0) and spread (1).

18. Measures of Position Definition
Percentiles – k th percentile is a set of data divides the lower k% from the upper (1-k)%
Divide into 100 parts, so 99 percentiles exist
“P sub k”
Use to give relative standing of the data

19. Measures of Position Definition
Quartiles – divides the data into four equal parts
Four parts, so three percentiles exist
“Q sub one, two, or three”
Q2 is the median of the data
Q1 is the median of the lower half
Q3 is the median of the upper half

20. Numerical summary of data
Five number summaries
Interquartile range (Q3 – Q1) is resistant to extreme values
Compute five number summary
Min value | Q1 | M | Q3 | max value

21. Building a Box Plot – part 1
1. Calculate interquartile range (IQR)
2. Compute lower & upper fence
Lower fence = Q1 – 1.5 (IQR)
Upper fence = Q (IQR)
3. Draw scale then mark Q1 and Q3
4. Box in Q1 to Q3 then mark M

22. Building a Box Plot – part 2
5. Temporarily mark fences with brackets
6. Draw line from Q1 to smallest value inside the lower fence and a line from Q3 to largest value inside the upper fence
7. Put * for all values outside of the fences
8. Erase brackets

23. Distribution based on Boxplot
Symmetric
median near center of box
horizontal lines about same length
Skewed Right / Positive Skew
median towards left of box
right line much longer than left line
Skewed Left / Negative Skew
median towards right of box
left line much longer than right line

24. Which measure best to report?
Symmetric distribution
Mean
Standard Deviation
Skewed distribution
Median
Interquartile Range

25. Self Quiz When can the mean and the median be about equal?
In the 2000 census conducted by the U.S. Census Bureau, two average household incomes were reported: $41,349 and $55,263. One of these averages is the mean and the other is the median. Which is which and why?

26. Self Quiz
The U.S. Department of Housing and Urban Development (HUD) uses the median to report the average price of a home in the United States.
Why do they do that?

27. Self Quiz
A histogram of a set of data indicates that the distribution of the data is skewed right.
Which measure of central tendency will be larger, the mean or the median?
Why?

28. Self Quiz _____ is a descriptive measure of a population
If a data set contains 10,000 values arranged in increasing order, where is the median located?
Matching: (parameter; statistic)
_____ is a descriptive measure of a population
_____ is a descriptive measure of a sample.

29. Self Quiz
A data set will always have exactly one mode. (true or false)
If the number of observations, n, is odd; then the median, M, is the value calculated by the formula M=(n+1)/2

30. Self Quiz
Find the Sample Mean:
20, 13, 4, 8, 10
83, 65, 91, 87, 84
Find the Population Mean:
3, 6, 10, 12, 14

31. Self Quiz The median for the given list of six data values is 26.5.
7 , 12 , 21 , , 41 , 50
What is the missing value?

32. Self Quiz
The following data represent the monthly cell phone bill for the cell phone for six randomly selected months.
$ $ $39.43
$ $ $49.26
Compute the mean, median, and mode cell phone bill.

33. Self Quiz
Heather and Bill go to the store to purchase nuts, but can not decide among peanuts, cashews, or almonds. They agree to create a mix. They bought 2.5 pounds of peanuts for $1.30 per pound, 4 pounds of cashews for $4.50 per pound, and 2 pounds of almonds for $3.75 per pound. Determine the price per pound of the mix.