Welcome to MDM4U (Mathematics of Data Management, University Preparation)

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1.1 Displaying Data Visually

Learning goal: Classify data by typeCreate appropriate graphs

MSIP / Home Learning: p. 11 #2, 3ab, 4, 7, 8

Why do we collect data? We learn by observing Collecting data is a systematic method of

making observations Allows others to repeat our observations

Good definitions for this chapter at: http://www.stats.gla.ac.uk/steps/glossary/alphabet.html

Types of Data 1) Quantitative – can be represented by a number

a) Discrete Data Data where a fraction/decimal is impossible E.g., Age, Number of siblings, Shoe size

b) Continuous Data Data where fractions/decimals are possible E.g., Weight, Height, Academic average

2) Qualitative – cannot be measured numerically E.g. Eye colour, Surname, Favourite band

Who do we collect data from? Population - the entire group from which we can

collect data / draw conclusions Data does NOT have to be collected from every member!

Census – data collected from every member of the pop’n Data is representative of the population Can be time-consuming and/or expensive

Sample - data collected from some members of the pop’n (min. 10%) A good sample will be representative of the pop’n Sampling methods in Ch 2

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 Number of absences

Monday 5

Tuesday 4

Wednesday 2

Thursday 0

Friday 8

Organizing Data (cont’d) Another useful organizer is a

stem and leaf plot. This table represents the

following data:101 103 107112 114 115 115121 123 125 127 127133 134 134 136 137 138141 144 146 146 146152 152 154 159165 167 168

Stem(first 2 digits)

Leaf(last digit)

10 1 3 7

11 2 4 5 5

12 1 3 5 7 7

13 3 4 4 6 7 8

14 1 4 6 6 6

15 2 2 4 9

16 5 7 8

Organizing Data (cont’d) What type of data is this? The class interval is the size of

the grouping, and is 10 units here 100-109, 110-119, 120-129, etc. No decimals req’d

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 2 4 5 5

12 1 3 5 7 7

13 3 4 4 6 7 8

14 1 4 6 6 6

15 2 2 4 9

16 5 7 8

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

Displaying Data – Bar Graphs Typically used for

qualitative/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

Bar graphs (cont’d) Double bar graph

Compares 2 sets of data

Internet use at Redwood Secondary School, by sex, 1995 to 2002

Stacked bar graph Compares 2 variables Can be scaled to 100%

Displaying Data - Histograms

Typically used for Continuous data

The bars are attached because the x-axis represents intervals

Choice of class interval size (bin width) is important. Why?

Want 5-6 intervals

Displaying Data –Pie / Circle Graphs A circle divided up

to represent the data

Shows each category as a % of the whole

See p. 8 of the text for an example of creating these by hand

Scatter Plot

Shows the relationship (correlation) between two numeric variables

May show a positive, negative or no correlation

Can be modeled by a line or curve of best fit (regression)

Line Graph

Shows long-term trends over time e.g. stock price, price of goods, currency

Box and Whisker Plot

Shows the spread of data Divided into 4 quartiles

Each shows 25% of the data Do not have to be the same size

Based on medians of entire data set, lower and upper half

See p. 9 for instructions

MSIP / Home Learning

p. 11 #2, 3ab, 4, 7, 8

Mystery Data

Gas prices in the GTA

3-Jan-08

22-Feb-08

12-Apr-08

1-Jun-08

21-Jul-0

8

9-Sep-08

29-Oct-

080.0000.2000.4000.6000.8001.0001.2001.4001.600

f(x) = − 1.78984476996036E-05 x² + 1.41853083716074 x − 28104.9051549717R² = 0.818508472651409

Hint: These values should get you pumped!

An example… these are prices for Internet service packages find the mean, median and mode State the type of data create a suitable frequency table, stem and leaf plot

and graph13.60 15.60 17.20 16.00 17.50 18.60 18.7012.20 18.60 15.70 15.30 13.00 16.40 14.3018.10 18.60 17.60 18.40 19.30 15.60 17.1018.30 15.20 15.70 17.20 18.10 18.40 12.0016.40 15.60

Answers… Mean = 494.30/30 = 16.48 Median = average of 15th and 16th numbers Median = (16.40 + 17.10)/2 = 16.75 Mode = 15.60 and 18.60 Decimals so quantitative and continuous. Given this, a histogram is appropriate

1.2 Conclusions and Issues in Two Variable Data

Learning goal:Draw conclusions from two-variable graphsDue now: p. 11 #2, 3ab, 4, 7, 8MSIP / Home LearningRead pp. 16–19Complete p. 20–24 #1, 4, 9, 11, 14

“Having the data is not enough. [You] have to show it in ways people both enjoy and understand.”- Hans Rosling

What conclusions are possible? To draw a conclusion, a number of conditions

must apply Data must address the question Data must represent the population

Census, or representative sample

Types of statistical relationships 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

Causation a change in one variable is PROVEN to cause a change in

the other requires an in-depth study e.g., incidence of cancer among smokers WE WILL NOT DO THIS IN THIS COURSE!!!

Case Study – Opinions of school 1 046 students were surveyed The variables were gender, attitude towards

school and performance at school.

Example 1 – Do female students like school more than male students do?

Example 1 – cont’d

The majority of females responded that they like school “quite a bit” or “very much”

Around half the males responded that they like school “a bit” or less

3 times as many males as females responded that they hate school

Since they responded more favorably, the females in this study like school more than males do

Example 2 – Is there a correlation between attitude and performance? Larger version on next slide…

Example 2 – cont’d Most students answered “Very well” when asked

how well they were doing in school. There is only one student who selected “Poorly”

when asked how well she was doing in school. Of the four students who answered “I hate

school,” one claimed he was doing well. It appears that performance correlates with

attitude Is 27 out of 1 046 students enough to make a

valid inference?

Example 3 – Examine all 1046 students

Example 3 - cont’d From the data, the following conclusions can be made: All students who responded “Very poorly” also

responded “I hate school” or “I don’t like school very much.”

A larger proportion of students who responded “Poorly” also responded “I hate school” or “I don’t like school very much.

It appears that there is a relationship between attitude and performance.

It CANNOT be said that attitude CAUSES performance, or performance CAUSES attitude without an in-depth study.

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?

01020304050

Yes No

Students Interested in Student Government

FemaleMale