1. Statistics: The Interpretation of Data
13.1 Organizing and Representing Data
13.2 Measuring the Center and Variation of
Data
13.3 Statistical Inference

2. 13.1
Organizing and Representing Data

3. VISUAL REPRESENTATIONS OF DATA
DOT PLOTS – used to summarize relatively small sets of data
From a
table -
To a
dot plot -

4. VISUAL REPRESENTATIONS OF DATA
STEM AND LEAF PLOTS – also used to summarize relatively small sets of data
From a
table -
To a stem
and leaf
plot -

5. VISUAL REPRESENTATIONS OF DATA
HISTOGRAMS – data is grouped into intervals

6. VISUAL REPRESENTATIONS OF DATA
LINE GRAPHS – also called frequency polygons

7. VISUAL REPRESENTATIONS OF DATA
BAR GRAPHS – used with categorical data, where the horizontal scale may be some nonnumerical attribute

8. VISUAL REPRESENTATIONS OF DATA
PIE CHARTS – represents relative amounts to a whole
Percent of each tax dollar expended by Mile High School District by category

9. VISUAL REPRESENTATIONS OF DATA
PICTOGRAPHS – useful in comparing quantities

10. 13.2
Measuring the Center and Variation of Data

11. MEASURES OF CENTRAL TENDENCY
MEAN – the arithmetic mean, or average
MEDIAN – the middle value in a
collection when the values are
arranged in order of increasing
size
MODE – the value that occurs most
frequently in a collection of
values

12. THE MEAN
The mean, or average, of a collection of values is
where S is the sum of the values and n is the number of values.

13. THE MEAN
A visual understanding using a data set of 7, 5, 7, 3, 8, and 6:

14. THE MEDIAN
Let a collection of n data values be written in order of increasing size.
If n is odd, the median, denoted by ,
is the middle value in the list.
If n is even, is the average of the two
middle values.

15. THE MEDIAN Data set 1: 24, 25, 25, 27, 29, 31, 32, 34, 37 Data set 2:
42, 42, 43, 44, 44, 46, 47, 47, 47, 49
average

16. THE MODE
A mode of a collection of values is a value that occurs the most frequently.
If two or more values occur equally often and more frequently than all other values, there are two or more modes.
Data Set:
42, 42, 43, 44, 44, 46, 47, 47, 47, 49
The mode of this data set is 47.

17. MEASURES OF VARIABILITY
RANGE – the difference between the
smallest and largest data values
QUARTILES – casually speaking, these
values divide the data set into four
sections, each of which contains, in
increasing order, about ¼ of the data
STANDARD DEVIATION – a measure of
the typical deviation from the mean

18. DEFINITION: OUTLIER
An outlier is a value that "lies outside" (is much smaller or larger than) most of the other values in a set of data.
Eg. Wayne Gretzky’s statistics

19. DEFINITION: STANDARD DEVIATION
Standard deviation is a measure of how spread out numbers are around the mean.

20. DEFINITION: STANDARD DEVIATION
Let be the values in a set of data and let denote their mean. Then
is the standard deviation.

21. Example 13.11 Computing a Standard Deviation
Compute the mean and standard deviation for this set of data:

22. 13.3
Statistical Inference

23. TERMINOLOGY AND NOTATION
A population is a particular set of objects about which one desires information.
Mean of a population =
Standard deviation of a population =
A sample is a subset of the population.
Mean of a sample =
Standard deviation of a sample =

24. DEFINITION: A RANDOM SAMPLE
A random sample of size r is a subset of r individuals from the population chosen in such a way that every such subset has an equal chance of being chosen.

25. THE NORMAL DISTRIBUTION

26. THE 68-95-99.7 RULE FOR NORMAL DISTRIBUTIONS
For a population that has a normal distribution, about 68% falls within 1 standard deviation of the mean, about 95% falls within 2 standard deviations of the mean, and about 99.7% falls within 3 standard deviations.

27. THE STANDARDIZED NORMAL CURVE

28. DEFINITION: PERCENTILE
A number such that the r-th percentage of a sample or distribution is less than or equal to that number is called the r-th percentile.
NOTE: Scoring at the 75th percentile on a test indicates that 75% of the students had a score less than or equal to yours, not necessarily that you got 75% of the problems correct.