1. Graphical Displays of Information
Chapter 3.1 – Tools for Analyzing Data
Mathematics of Data Management (Nelson)
MDM 4U

2. Histograms
contain continuous data grouped in class intervals, which will display how data is spread over a range
the width of each bar is known as the bin width
different bin widths produce different shaped distributions
bin widths should be equal and there should be at least five (5)

3. Histogram Example these histograms represent the same data
however, one shows much less of the structure of the data
too many bins (bin width too small) is also a problem

4. Histogram Applet – Old Faithful

5. Bin Width Calculation
the bin width is calculated by dividing the range = max – min by the number of intervals you desire (5-6)
the bins should not overlap
wrong: 0-10, 10-20, 20-30, 30-40
Discrete
correct: 0-10, 11-20, 21-30, 31-40
Continuous
correct: , , ,

6. Mound-shaped distribution
The middle interval(s) have the greatest frequency (i.e. the tallest bar)
The bars get smaller as you move out to the edges.

7. U-shaped distribution
Lowest frequency in the centre, highest towards the outside
E.g. height of a combined grade 1 and 6 class

8. Uniform distribution All bars are approximately the same height
E.g. roll a die 50 times

9. Symmetric distribution
A distribution that is the same on either side of the mode (tallest bar)
U-Shaped, Uniform and Normal Distributions

10. Skewed distribution (left and right)
Highest frequencies at one end
E.g. the years on a handful of quarters

11. Exercises Define in your notes:
Frequency distribution (p. 146)
Cumulative frequency (p. 146)
Relative frequency (p. 146)
Try page 146 #1,2,3, 11 (use Excel or Fathom),13

12. Measures of Central Tendency
Chapter 3.2 – Tools for Analyzing Data
Mathematics of Data Management (Nelson)
MDM 4U

13. Sigma Notation
the sigma notation is used to compactly express a mathematical series
ex: … + 15
this can be expressed:
the variable k is called the index of summation.
the number 1 is the lower limit and the number 15 is the upper limit
we would say: “the sum of k for k = 1 to k = 15

14. Examples: write in expanded form:
= [2(4) + 1] + [2(5) + 1] + [2(6) + 1] + [2(7) + 1] =
=48
note that any letter can be used for the index of summation, though k, a, n, i, j & x are often used

15. Example: write the following in sigma notation

16. The Mean
found by dividing the sum of all the data points by the number of elements of data
Deviation
the distance of a data point from the mean
calculated by subtracting the mean from the value

17. The Weighted Mean
where xi represent the data points, wi represents the weight or the frequency
see examples on page 153 and 154
example: 7 students have a mark of 70 and 10 students have a mark of 80
mean = (70 * * 10) / (7 + 10)

18. Means with grouped data
for data that is already grouped into class intervals (assuming you do not have the original data), you must use the midpoint of each class to estimate the weighted mean
see the example on page 154-5

19. Median the midpoint of the data
calculated by placing all the values in order
if there are an even number of values, the median is the mean of the middle two numbers
median = 7
if there is an odd number of values, the median is the middle number
median = 6

20. Mode Simply chosen by finding the number that occurs most often
There may be no mode, one mode, two modes (bimodal), etc.
Which distributions from yesterday have one mode?
Mound-shaped, Left/Right-Skewed
Two modes?
U-Shaped, some Symmetric
Multiple modes?
Uniform
Modes are appropriate for discrete data or non-numerical data
shoe sizes
shoe colors

21. Distributions and Central Tendancy
the relationship between the three measures changes depending on the spread of the data
symmetric (mound shaped)
mean = median = mode
right skewed
mean > median > mode
left skewed
mean < median < mode

22. What Method is Most Appropriate?
Outliers are data points that are quite different from the other points
Outliers have the greatest effect on the mean
Median is least affected by outliers
Skewed data is best represented by the median
If symmetric either median or mean
If not numeric or if the frequency is the most critical, use the mode

23. Example 1 find the mean, median and mode
mean = [(1x2) + (2x8) + (3x14) + (4x3)] / 27 = 2.7
median = 3
mode = 3
which way is it skewed?
Left
Survey responses 1 2 3 4
Frequency 8 14

24. Example 2 Find the mean, median and mode
mean = [(145x3) + (155x7) + (165x4)] / 14 = 155.7
median = 155
mode =
which way is it skewed?
Mound-shaped
Height
No. of Students 3 7 4

25. Exercises try page 159 #4, 5, 6, 8 Remembrance Day by the Numbers

26. References
Wikipedia (2004). Online Encyclopedia. Retrieved September 1, 2004 from