1. Prepared by: C.Cichanowicz, March 2011
Statistics
Prepared by: C.Cichanowicz, March 2011

2. Measures of Central Tendency
Ungrouped Data
Measures of Central Tendency
Prepared by: C.Cichanowicz, March 2011

3. Prepared by: C.Cichanowicz, March 2011
Mode (MOD)
The data value with the largest frequency (the one that appears the most often).
A data distribution can have more than one mode, one mode, ore no mode.
The mode is representative of the data when there is a data value with a large frequency.
Prepared by: C.Cichanowicz, March 2011

4. Prepared by: C.Cichanowicz, March 2011
Median (MED)
The data value in the middle of the distribution.
Arrange the data in increasing order.
Odd # of data: the median is the value in the middle.
Even # of data: there will be 2 values in the middle, take their average.
The median is representative of the data when the data values are far from each other.
Prepared by: C.Cichanowicz, March 2011

5. Prepared by: C.Cichanowicz, March 2011
Mean
The average of the data.
Add all the data values together and divide the sum by the total number of values in the distribution.
The mean is representative when the data values are close together
Prepared by: C.Cichanowicz, March 2011

6. Box-and-Whisker Plots
Prepared by: C.Cichanowicz, March 2011

7. Box-and-Whisker Plots
A box-and-whisker plot is made up of 5 values.
The minimum
The maximum
3 quartiles
Q2: median of the distribution
Q1 :median of the first half of the distribution
Q3 :median of the second half of the distribution
Prepared by: C.Cichanowicz, March 2011

8. Box-and-Whisker Plots
Procedure
Arrange the data in increasing order.
Determine the median (Q2).
Determine the median of the first half of the data (Q1).
Determine the median of the second half of the data (Q3).
Draw a number line, with even spacing between numbers. Remember to consider the range of your data.
Mark with vertical lines the 5 values. (min, Q1, Q2, Q3, max)
Draw a box that connects Q1, Q2, and Q3.
Draw a line that connects min to Q1 and Q3 to max (these are the whiskers)
Prepared by: C.Cichanowicz, March 2011

9. Box-and-Whisker Plots
Example:
40, 86, 32, 66, 87, 76, 32, 45
32, 32, 40, 45, 66, 76, 86, 87 Q2 Q1 Q3
min Q1 Q2 Q3
max
Prepared by: C.Cichanowicz, March 2011

10. Box-and-Whisker Plots
The box-and-whisker plot separates the data into 4 equal parts (quartiles) ... There is 25% of the data in each part, even though it may not look like it
min Q1 Q2 Q3
max
25% of data
Prepared by: C.Cichanowicz, March 2011

11. Measures of Dispersion
Range = max – min
Interquartile Range = Q3 – Q1
Outliers: data values that are numerically distant from the others.
There is an outlier if a whisker is  1.5 times the length of a box (in the box-and-whisker plot).
Prepared by: C.Cichanowicz, March 2011

12. Distribution Tables and Histograms
Prepared by: C.Cichanowicz, March 2011

13. Prepared by: C.Cichanowicz, March 2011
Distribution Tables
Vocabulary:
Range of a distribution = max value – min value
Frequency of a class = the number of data that belong to that class
Relative frequency = ratio of the frequency of a class to the total number of data (percentage)
Prepared by: C.Cichanowicz, March 2011

14. Prepared by: C.Cichanowicz, March 2011
Distribution Tables
class
frequency
Relative frequency
[ , [
To make a distribution table:
Determine the range of the data.
Divide the range into the desired number of classes (5-10, depending on the size of the data).
Each class must be the same size.
Classes must cover entire range without overlapping.
Fill in the table, with the classes in order, and determine the number of data in each class.
Prepared by: C.Cichanowicz, March 2011

15. Relative frequency (%)
Histograms
Use the distribution table you make to prepare a histogram. 5 15 10 20 25 30
class
Frequency
Relative frequency (%)
[5, 10[ 5 10
[10, 15[ 9 18
[15, 20[ 16 32
[20, 25[ 13 26
[25, 30[ 7 14
Total 50
100
Prepared by: C.Cichanowicz, March 2011

16. Measures of Central Tendency
Grouped Data
Measures of Central Tendency
Prepared by: C.Cichanowicz, March 2011

17. Relative Frequency (%)
Modal Class
The modal class is the class with the largest frequency.
The mode is the middle value of the modal class.
Class
Frequency
Relative Frequency (%)
[50, 60[ 2 8
[60, 70[ 4 16
[70, 80[ 10 40
[80, 90[ 7 28
[90, 100[
Total 25
100
The class with the highest frequency or relative frequency is [70, 80[, so that is the modal class. The mode is 75.
Prepared by: C.Cichanowicz, March 2011

18. Relative Frequency (%) Cumulative Frequency (%)
Median
The median is in the class where 50% of the data falls. The median is the middle value of that class.
Add a separate column to the distribution table...the cumulative frequency.
Class
Frequency
Relative Frequency (%)
Cumulative Frequency (%)
[50, 60[ 2 8
[60, 70[ 4 16 24
[70, 80[ 10 40 64
[80, 90[ 7 28 92
[90, 100[
100
Total 25
50% of the data lies in the class [70, 80[, so the median is 75.
Prepared by: C.Cichanowicz, March 2011

19. Using relative frequency
Mean
Determine the middle of each class.
Multiply the middle value of each class by the frequency of that class.
Calculate the sum.
Divide the sum by the total number of data values.
Determine the middle of each class.
Multiply the middle of each by the relative frequency (the percentage turned into a decimal).
Add up the values obtained, they will total the mean.
Method 1
Using frequency
Method 2
Using relative frequency
Prepared by: C.Cichanowicz, March 2011

20. Mean Class Frequency Relative Frequency (%) Cumulative Frequency (%)
Middle value of class
Mean using frequency
Mean using relative frequency
[50, 60[ 2 8 55
55*2=110
4.4
[60, 70[ 4 16 24 65
65*4=260
10.4
[70, 80[ 10 40 64 75
75*10=750 30
[80, 90[ 7 28 92 85
85*7=595
23.8
[90, 100[
100 95
95*2=190
7.6
Total 25
1905
76.2
1905/25=76.2
Prepared by: C.Cichanowicz, March 2011

21. Samples and Sampling Methods
Prepared by: C.Cichanowicz, March 2011

22. Choosing a representative Sample
Is the sample representative or not?
The sample must be representative of a target population. It must have as many characteristics as possible found in the target population.
Depends on the sample size and sampling method
Sources of bias
Sampling method used.
Sample not representative of the population.
A poorly formulated question.
Attitude of person conducting the survey.
Inadequate representation of the results.
Rejecting to large a portion of the sample.
Prepared by: C.Cichanowicz, March 2011

23. Prepared by: C.Cichanowicz, March 2011
Sampling Methods
Random Sampling
Each element is chosen at random.
Each element has an equal chance of being chosen.
Good when the population is homogeneous (elements have the same characteristics).
Prepared by: C.Cichanowicz, March 2011

24. Prepared by: C.Cichanowicz, March 2011
Sampling Methods
Systematic Sampling
Need a list of elements.
Each element is chosen at regular intervals
Prepared by: C.Cichanowicz, March 2011

25. Prepared by: C.Cichanowicz, March 2011
Sampling Methods
Stratified Sampling
Split population into subgroups (strata), made up of groups with the same characteristics.
Determine the percentage of the elements of each subgroup in relation to the total population (ex. 20% French, 75% English, 5% Spanish).
Elements are chosen at random from each subgroup, keeping in mind the percentages that were determined for each subgroup (ex. If a total of 20 people are to be chosen, take 4 French, 15 English, and 1 Spanish).
Prepared by: C.Cichanowicz, March 2011

26. Prepared by: C.Cichanowicz, March 2011
Sampling Methods
Cluster Sampling
It’s the subgroups of a population that are studied.
Subgroups are chosen at random (clusters).
All elements within the clusters make up the sample.
Prepared by: C.Cichanowicz, March 2011

27. Prepared by: C.Cichanowicz, March 2011
Statistical Graphs
Prepared by: C.Cichanowicz, March 2011

28. Prepared by: C.Cichanowicz, March 2011
Qualitative Data
Bar graph
Vertical or horizontal bars
Used to compare qualities
Circle Graph
Each sector represents a category
Displays percentages of a whole
Prepared by: C.Cichanowicz, March 2011

29. Prepared by: C.Cichanowicz, March 2011
Quantitative Data
Broken-line graph
Used to represent chronological data, data that changes over time.
Scatter plot
Used to see if there is a link between two aspects of a population.
Each point represents a element.
Prepared by: C.Cichanowicz, March 2011

30. Prepared by: C.Cichanowicz, March 2011
Quantitative Data
Histogram
Represents distributions of continuous data, grouped data.
Provides an overview of the distribution.
Box-and-whisker plot
Provides an overview of a distribution.
4 groups with 25% of the data in each group.
We can see if data is symmetrical.
Prepared by: C.Cichanowicz, March 2011