Transcript Mu123week5

Discovering Mathematics
Week 5
BOOK A - Unit 4:
Statistical Summaries
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Unit 4: Statistical Summaries
This unit includes the following sections:
• Questions
• Dealing with data
• Summarising data: Location
• Summarising data: Spread
• Measuring with accuracy and precision
• Statistical Pictures (Unit 11)
• Discussing TMA
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1. Questions
This unit deals with statistical thinking. It starts with
questions that can be addressed by statistical methods.
Statistical information is all around you and arises as an
attempt to answer questions of various kinds. (e.g.
Should people stop smoking?)
Statistical thinking is presented as a helpful way of
seeing the world quantitatively. For examples: table of
numbers, graphs & charts.
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Section 0: What is Statistics?
Application Areas
• Economics
– Forecasting
– Demographics
• Sports
– Individual & Team
Performance
• Engineering
– Construction
– Materials
• Business
– Consumer Preferences
– Financial Trends
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Types of statistical questions
• Most of the questions of investigations can
be categorized into one of the following
types :
- Summarising: how can the information be
reduced.
- Comparing: is there a difference?
- Seeking a relationship: what sort of
relations is there.
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Types of statistical questions(II)
• Summarizing questions:
Summarizing information is by reducing the many figures to just one
representative number or using charts or plots. Examples of
investigations:
- How many people die from road accidents each day.
- How old are the students studying MU123
• Comparing questions:
Comparing investigation is to decide whether or not there is a
difference between two things or comparing between two averages.
Examples are:
- Do more people, on average, die from road accidents on
weekdays or at weekends?
- Are students studying MU123 older or younger than students on
LB160?
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Types of statistical questions(III)
• Relationship Questions:
Investigations could be a relationship between quite separate
things. What the relationship is. For example:
- As the numbers of road deaths in different countries linked
to their respective speed limits?
- What is the connection between the numbers of hours that
students work on a level 3 course in mathematics and their
final grade.
Discuss in the class Activities I and 2 (page 179) and distinguish
between the three types of statistical investigations.
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The statistical investigation cycle
There are four stages in most statistical investigations:
P: Posing a question. This is the start of the investigations.
C: Collecting data, choosing samples or design a survey.
A: Analyze the data, calculating averages and plotting graphs
I: Interpreting the results, takes the action back and see whether
the data analysis help to answer the posed question.
(check Activities 3 and 4 for complete example)
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Activity
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Dealing with Data
Data that you collect yourself are called primary data.
Data that already exist and can be used or adapted are called
Secondary data.
It is extremely important when presenting any dataset, whether
primary or secondary, is to provide an accurate reference to the
data source.
Take a quick look at the dataset at page 184. Explain the rows and
columns. Solve Activity 6.
Discrete data are data that can take one of a particular set of
values. E.g: Number of days in a week you study, or number of
websites you visit per day.
Binary data: are the data coded by two numerical values. 1,0
Solve activity 9
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Summarising data: location
Statisticians call the location of a dataset to represent a single
number that represents an ‘average’, typical or ‘central’ value.
It is always good idea to scan the data, looking closely at the
numbers to see if there is a missing data, inspect the data.
Measuring Location:
The most important summary of a dataset is a measure of its
location. The three most common measures of location are:
Mean, the mode and the median.
Let us look at the following examples:
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Summarising data: location (II)
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Summarising data: location (II)
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Summarising data: Spread
A second basic property of data is how widely spread the values are.
Range:
TMA marks for six students are: 42, 58, 60, 68, 78, 92
42 in MIN and 92 is MAX , this show the data is wide spread.
Another TMA marks are: 60, 65, 72, 74, 75, 80 this ranges from 60 to
80,
this is much narrower.
Range is the difference between the two values Maximum and
Minimum after arranging the dataset in increasing order.
Range = Max – Min = 80-60 = 20
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Summarising data: Spread
Quartiles:
Lower quartile (Q1) and upper quartile (Q3)
The Median Q2 is (72+74)/2 = 73
Lower quartile = 65
Upper quartile = 75
Interquartile range = 75 – 65 = 10
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Standard Deviation
• Standard deviation (SD) is the best known measure of spread.
• To find the SD, follow the following steps:
Assume that we have the following data: 1,2,4,6,7
Solve Activity 26
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Summary
• Location and spread are the two key forms of
data summary, that two measures of location
are the mean and median, and that three
measures of spread are the range, the
interquartile range and the standard
deviation.
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Unit 11. Statistical pictures
Section 2: Histograms, bar charts, pie charts
2.1 Histogram (Continuous data)
Exercise:
consider the dataset below, which consists of the weights in
kilograms of 32 women at the start of their pregnancies.
54.5 59.1 73.2 41.4 55.5 70.5 76.4 70 52.3
83.2 64.5 49.5 70
52.3 68.2 47.3 66.8 70
56.4 53.6 59.1 44.5 55.9 57.3 69.5 73.2 62.7
73.6 70
56.7 58.2 68.2
A class is one of the
categories
The class
frequency is the
number of
observations in a
particular class.
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Section 2: Histograms, bar charts, pie charts
2.2 Bar charts (Discrete data)
Exercise:
The bar chart below shows the frequencies with which the
six scores 1, 2, 3, 4, 5, 6 cropped up as a result of 30 rolls of
a die.
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Section 2: Histograms, bar charts, pie charts
2.2 Bar charts (Categorical data)
Exercise:
Major
Accounting
Economics
Management
Total
Count
130
20
50
200
150
100
50
0
Acct.
Econ.
Mgmt.
Major
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Section 2: Histograms, bar charts, pie charts
2.4 Pie Charts (Categorical data)
1. Shows breakdown of total quantity into
categories
2. Useful for showing relative differences
3. Angle size =(360°)(percent)
Mgmt.
25%
Econ.
10% 36°
Major
%
Angle
Accounting
65
234
Economics
10
36
Management
25
90
Total
100
360
Acct.
65%
Revision :
The four stages of a statistical investigation cycle PCAI
Stage 1 Pose a question
Stage 2 Collect relevant data
Stage 3 Analyse the data
Stage 4 Interpret the results
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Revision:
Recall Activity:
Here are nine of the common types of task that tend to arise
in the C, A and I stages of a statistical investigation. Try to
match each task to one of these three stages
1. Calculate an average
2. Calculate a percentage
3. Choose a set of values, or sample
4. Make a decision based on an observed, numerical
difference
5. Design a questionnaire
6. Draw a conclusion
7. Draw a helpful graph
8. Key the data into a spreadsheet
9. Make a prediction about the real world
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Discussing the TMA
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Thank you