1. Displaying Data Visually
Chapter 1.1 – The Power of Information
Mathematics of Data Management (Nelson)
MDM 4U

2. Why do we collect data? We learn by observing
Difficult to make accurate decisions if poor observations made
Collecting data is a systematic method of making observations
Helps to yield representative data
Allows others to repeat our observations
Good definitions for this chapter at:

3. Scales of Measurement of Data
Qualitative data:
1. Nominal Data
no order - gives names or labels to categories
e.g., Gender, Eye colour, Marital status

4. Types of Data 1) Quantitative – can be represented by a number
2) Qualitative – cannot be represented by a number (meaningfully)
E.g. Hair colour 1 = black, 2 = brown, 3 = blonde, 4 = red

5. Scales of Measurement of Data (cont’d)
Quantitative Data:
2. Ordinal Data
Has order, but the interval between measurements is not meaningful
e.g., Level of Education, Likert Scale attitude statements:
1 – strongly disagree
2 - somewhat disagree
3 - neither agree nor disagree
4 - somewhat agree
5 - strongly agree

6. Scales of Measurement of Data (cont’d)
3. Interval Data
Has order and equal intervals between levels
No true starting point (zero)
E.g., Celsius temperature, Tide height, Longitude
4. Ratio Data
The highest level of measurement
Has order, equal intervals, absolute starting point
E.g., Weight, Height, Kelvin Temperature (absolute zero = °C, [K] = [°C] )
(Caulkins, 2003)

7. Quantitative Data Discrete Data
Data where a fraction/decimal is impossible
E.g., Age, Shoe size, Number of children, Number of courses
Continuous Data
Data where fractions/decimals are possible
E.g., weight, height, academic average
Nominal and Ordinal measurements generally give rise to discrete data
Interval and Ratio measurements can give rise to either type (James Cook University, n.d.)

8. Who do we collect data from?
Population - the entire group from which we can collect data
NOTE: Data does NOT have to be collected from every member
Census – data is collected from every member of the pop’n
This may be too time-consuming or expensive
Sample - data is collected from some members of the pop’n (10-20%)
A good sample must be representative of the pop’n
next unit

9. Organizing Data
A frequency table is often used to display data, listing the variable and the frequency.
What type of data does this table contain?
Intervals can’t overlap
Use from 3-12 intervals / categories
Day
Frequency
(number of absences)
Monday 3
Tuesday 5
Wednesday 6
Thursday 4
Friday 2

10. Organizing Data (cont’d)
Stem
(first 2 digits)
Leaf
(last digit) 10
1 3 7 11 12 13 14 15 16
5 7 8
Another useful organizer is a stem and leaf plot.
This table represents the
following data:

11. Organizing Data (cont’d)
What type of data is this?
The class interval is the size of the grouping, and is 10 units here
, , , etc.
What is the effect of reducing or increasing the interval size?
Increase frequency gets larger
Decreasefrequency gets smaller
stem can have as many numbers as needed
a leaf must be recorded each time the number occurs
Stem
Leaf 10
1 3 7 11 12 13 14 15 16
5 7 8

12. Measures of Central Tendency
Used to indicate one value that best represents a group of values
Mean (Average)
Add all numbers and divide by the number of values
Affected greatly by outliers (values that are significantly different from the rest)
Median
Middle value
Place all values in order and choose middle number
For an even # of values, average the 2 middle ones
Not affected as much by outliers
Mode
Most common number
There can be none, one or many modes
Only choice for Qualitative data

13. Displaying Data – Bar Graphs
Typically used for nominal/discrete data
Shows how certain categories compare
Why are the bars separated?
Would it be incorrect if you didn’t separate them?
Number of police officers in Crimeville, 1993 to 2001

14. Bar graphs (cont’d) Double bar graph Stacked bar graph
Compares 2 sets of data
Stacked bar graph
Compares 2 variables
Internet use at Redwood Secondary School, by sex, 1995 to 2002

15. Displaying Data - Histograms
Typically used for Continuous data (Interval and Ratio)
The bars are attached because the x-axis represents intervals
Choice of class interval size is important. Why?

16. Displaying Data –Pie / Circle Graphs
A circle divided up to represent the data
Shows each category as a portion of the whole
See p. 8 of the text for an example of creating these by hand

17. Examining the spread of data
the box and whisker plot is one way to indicate the spread of data
divided into 4 quartiles (25% of the data in each)
e.g., 66, 68, 74, 77, 82, 88, 93, 94
instructions for creating these may be found on page 9 of the text or at:

18. Examining Trends A line graph shows long-term trends
e.g. stock price, moving average

19. MSIP / Homework
p. 11 #2, 3ab, 4, 7, 8

20. Mystery Data
Gas prices in the GTA

21. An example… these are prices for Internet service packages
find the mean, median and mode
determine the scale of measurement for the data
determine what type of data this is
create a suitable frequency table, stem and leaf plot and graph

22. Answers…
Mean = /30 = 16.48
Median = average of 15th and 16th numbers
Median = ( )/2 = 16.75
Mode = and 18.60
The data is numerical, so at least Interval data. It has an absolute starting point, so it is ratio data.
Decimals so quantitative and continuous.
Given this, a histogram is appropriate

23. Conclusions and Issues in One Variable Data
Chapter 1.2 – The Power of Information
Mathematics of Data Management (Nelson)
MDM 4U

24. What conclusions are possible?
To draw a conclusion, a number of conditions must apply
data must be representative of the population (achieved by random sampling)
sample size must be large enough
data must address the question

25. Types of statistical relationships
What is a correlation?
two variables appear to be related
i.e., a change in one variable is associated with a change in the other
e.g., salary increases as age increases
What is a causal relationship?
a change in one variable is proven to cause a change in the other
usually requires an in-depth study i.e. WE WILL NOT DO THIS IN THIS COURSE!!!
e.g., incidence of cancer among smokers
e.g., hours of study per week with the approach of exams

26. Drawing Conclusions
Do females seem more likely to be interested in student government?
Does gender appear to have an effect on interest in student government?
Is this a correlation?
Is it likely that being female causes interest?

27. MSIP / Homework
Read pp. 16 – 19
Complete p. 20 – 24 #1, 4, 9, 11, 14

28. References
Calkins, K. (2003). Definitions, Uses, Data Types, and Levels of Measurement. Retrieved August 23, 2004 from
James Cook University (n.d.). ICU Studies Online. Retrieved August 23, 2004 from