1. Math 201: Chapter 2
Sections 3,4,5,6,7,9

2. 2-3 Summation Notation

3. 2-4 Numerical Measures of Central Tendency
Data set usually refers to a sample (hardly ever a population!) We make inference about the measures of the population using the sample measures.
What are those measures?
These are some numerical descriptive measures.
Not graphical! We have seen graphical descriptions in Sections 2-1 and 2-2.

4. Numerical Descriptive Measures
The central tendency: the tendency of the data to cluster, or center, about certain numerical values
The variability: the spread of the data
Center
Spread

5. How do we measure?
The mean of a data set is the average of the numbers:
Mean MPG is average of 100 MPG values, found as
Population mean: versus Sample mean:
The median of a quantitative data set is the middle number when the data set is ordered:
Ex. Median of MPG measurements is 37.0

6. Median or Mean?
Which measure is better in identifying the center of the measurements (data)?
Mean is more prone to outliers! Median is better in such situations.
Ex. 5, 7, 4, 5, 20, 6, 2
Ex. Typical income of a community.
__________________________
Ex. 5, 7, 4, 5, 20, 6, median?

7. Income in US in 1992 Median: $36,800 Average: $44,500
A distribution with long right-hand tail
that is, skewed to the right.
One may use the median rather than the average, if the distribution is skewed (to the right or left) since the average pays too much attention to the extreme tail of the distribution.

8. Mean versus Median and Skewness
½ the area
Skewed to the right Symmetric Skewed to the left

9. 2-5 Numerical Measures of Variability
The range of a data set is the difference of the largest and smallest measurements.
A more refined measure of variability
Sample variance:
Ex. 1,2,3,4,5
Ex. 2,3,3,3, (Hint: Look at their dot diagrams!)

10. Standard Deviation Standard deviation is the square root of variance
Population variance is also conceived and denoted by and population standard deviation by .
Use of statistical (computer) packages are handy for larger data sets MINITAB, SAS.

11. 2.6 Interpreting Standard Deviation
Useful for comparing the variability of two different samples
Larger standard deviation more variability
What about a single sample? What is the meaning of standard deviation?
Chebyshev’s rule for any data set
Empirical rule for mound shaped data
We are after the percentage of observations that fall within one, two, etc. standard deviations of the mean.
These rules apply to population as well.

12. Ex. MPG data Recall that and s = 2.42.
We want to know the frequency (and percentage) of observations in the intervals
(34.57, 39.41) : within one standard deviation of the mean
(32.15,41.83) : within two standard deviations of the mean

13. Chebyshev’s Rule
It is possible that very few measurements fall within one standard deviation (s or ) of the mean ( or )
At least ¾ of the measurements will fall within two standard deviations
At least 8/9 of the measurements will fall within three standard deviations.
For any number k greater than 1, at least 1-1/k2 of the measurements will fall within k standard deviations of the mean

14. The Empirical Rule
For data sets with frequency distributions that are mound shaped and symmetric like
Approximately,
68% of the measurements fall into one standard deviation of the mean
95% of the measurements fall into two standard deviations of the mean
99.7% of the measurements fall into three standard deviations of the mean

15. Examples 1. RATMAZE MINITAB descriptive statistics
0.6
1.74
2.47
3.65
4.44
5.63
1.06
1.93
2.75
3.77
4.55
6.06
1.15
1.97
3.16
3.81
5.15
7.6
1.65
2.02
3.2
4.02
5.21
8.29
1.71
2.06
3.37
4.25
5.36
9.7
Examples
1. RATMAZE
MINITAB descriptive statistics
Variable N Mean StDev Minimum Median Maximum
RUNTIME
Check if the empirical rule applies…
2. Making a statistical inference
A manufacturer of automobile batteries claims that the average length of their life is 60 months. However, the guarantee is 36 months. Suppose the standard deviation is 10. …. See pg. 71.

16. 2.7 Relative Ranking: Percentiles and z-score
These are measures of relative standing.
For any set of n measurements, the pth percentile is a number such that p% of the measurements are at or below that number and (100-p)% fall at or above it.
A formula to find pth percentile:
Find p% of n, say k. If k is an integer, average kth and (k+1)st observations in the data set. If k is not an integer, than round it to the next larger integer and find the observation with that ranking.
Ex. Consider RATMAZE data

17. z-score For the sample or population: (data point – mean) / std. dev.
Ex. Suppose a sample of 2000 high school seniors’ verbal SAT scores is selected, which have and s = 75. Joe Smith’s score is 475. Find the z-score and interpret as relative standing.
Interpretation for mound shaped population…

18. 2.9 Graphing Bivariate Relationships
A scatterplot describes the relationship between two quantitative variables, a bivariate relationship.

19. Example: Cost vs Floor Area