Statistics & Probabilities

Grade 9 Pre IB

Data Types and Representation

Continuous Numerical DataIn this type a continuous range represents the

continuous values that the data can take

Discrete Numerical DataIn this kind a discrete range represents the

discrete values that the data can take

Continuous Numerical Data

A continuous numerical variable can theoretically take any value on the number line

Example: The weight of pumpkins harvested by Salvi in kilograms was recorded as:

2.1 ,3.0, 0.6, 1.5, 1.9, 2.4, 3.2, 4.2, 2.6, 3.1, 1.8, 1.7, 3.9, 2.4, 0.3, 1.5, 1.2

Continuous Numerical Data

The data is continuous because the weight could be any value from 0.1 kg up to 10 kg

The range of weights recorded is: 0.3 to 4.2

This data can be represented by the following graph

Continuous Numerical Data

Weight (Kg)

Frequency

0 < 1 21 < 2 62 < 3 43 < 4 44 < 5 1

0 < 1 1 < 2 2 < 3 3 < 4 4 < 50

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7

Discrete Numerical Data Data is made up of individual

observations of a variableDiscrete numerical variable can

only take distinct values which we find by counting

1. Consider the set of Math, Science, History, and English courses and collect the following data

2. How many of you like Math the best?3. How many of you like Science the best?4. How many of you like History the best?5. How many of you like English the best?6. Now we draw a graph representation of

the collected data on a chart:

Discrete Numerical Data Example

Discrete Numerical Data

Categorical Data To find differences between numerical and categorical data

answer the following questions: How many pets do students in our class have? How many hours a week do you spend watching TV? What is your favourite sport? What kind of music do you like best? How many hours a week do you talk on the phone? What kinds of snacks do you like? How much do our backpacks weigh? How much candy do we eat each week?

StatisticsIn today’s fast changing and moving

computer age we collect vast quantities of data

Math is concerned with how data is collected, organized, presented, summarized and then analyzed

StatisticsStatistics is the branch of mathematics

that deals with the collection, organization, and interpretation of data

These data are usually organized into tables and/or presented as graphs

Some common types of graphs are:

Statistics Showing by GraphsPictograph

A graph that uses a symbol or an image to represent a certain amount

In a circle graph a complete set of data is presented by the circle

Various parts of the data are represented by the sectors of the circle

This method is useful when showing data as a percentage or as a fraction of the entire data space is needed

Statistics Showing by GraphsCircle Graph

Statistics Showing by GraphsCircle Graph

Example of Circle GraphMath grade 9 Mark

distribution

Knowledge 30%

Thinking 10%

Application 12%

Communication 13%

Mid Year Exam 5%

Final Exam 10%

Calculator Test 10%

EQAO Testing 10%

The graph uses vertical bars to represent different segments of the data, and it is used for discrete data

As an example the students favorite fruit juice color can be represented by a bar graph

Statistics Showing by GraphsBar Graph

A graph that uses bars to represent the frequency (or number) of the data within a range of values

It is used for continuous dataAs an example the distribution of salaries

of employees in a company can be represented by a histogram

Statistics Showing by GraphsHistogram Graph

Statistics Showing by GraphsHistogram Graph

Plots different data values on the y-axis

The only points on a broken-line graph that represent data are the endpoints of the segment

The adjacent points are joined by a line segment

The exact value is not clear between the points

Statistics Showing by GraphsBroken-Line Graph

Statistics Showing by GraphsBroken-Line Graph

As an example, mass of a rabbit at different months of the year is plotted

This graph shows the value of one variable corresponding to the value of another variable for all values over a given interval

All the points on a continuous-line graph correspond to data

As an example the following graphs shows the distance required to bring a car to rest from the moment the brakes are applied versus the car speed up to 100 km/h

Statistics Showing by GraphsContinuous-Line Graph

Statistics Showing by GraphsContinuous-Line Graph

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Speed (k/h)

Stopping Distance (m)

Using Scatter PlotScatter plot is a graphic tool used to display the

relationship between two quantitative variablesA scatterplot uses X and Y axis and series of

dotsEach dot represents one observation from a data

pointThe position of the dots on the scatter plot

represents its X and Y values

Scatter Plot ExampleWeight versus Height of Basketball Players

Height (Inches) weigh (Pounds)67

15572

22077

24074

19569

175

Interpreting the plotEach player is represented by a dot

on the scatter plotThe first dot represents the shortest

playerThis plot suggests that relationship

between height and weight can be approximately modeled by a linear line with a positive slope

Activity: Scatter Plot1. Measure your hand span2. Measure your height3. Gather the data from all members of

your class and put it in a table 4. Choose one variable as the independent

variable and the other as the dependent variable, draw a scatter plot to represent the data and then in a few sentences interpret your data!

Scatter Plot1. Would you say the variables are

continuous or discrete?2. Are there any data points that don’t

fit the pattern? If so, explain.3. How does the scatter plot suggest

how hand span and height are related?

Class Arm-Span vs Height

100 110 120 130 140 150 160 170 180 190 200100

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Height

HeightLinear (Height)

Stem-and-Leaf PlotThis kind of plot is used in organizing a discrete

numerical dataAll of the actual data values are shownThe minimum or smallest data value is easy to

findThe maximum or largest data value is easy to

findThe range of values that occurs most often is

easy to seeThe shape if the distribution of the data is easy

to see

Exercise for the Stem-Leaf Plot

The score for test out of 50 was recorded for 36 students:

25, 36, 38, 49, 23, 46, 47, 15, 28, 38, 34, 9, 30, 24, 27, 27, 42, 16, 28, 31, 24, 46, 25, 31, 37, 35, 32, 39, 43, 40, 50, 47, 29, 36, 35, 33

Organize the data using a stem-and-leaf plotWhat percentage of students scored 40 or more

marks?

The Solutions and Plot The stem will be 0, 1, 2, 3, 4, 5 Unordered Stem-Plot

Stem Leaf0 91 5 62 5 3 8 4 7 7 8 4 5 93 6 8 8 4 0 1 1 7 5 2 9 6

5 34 9 6 7 2 6 3 0 75 0

Stem-and Leaf Plot Ordered Stem-plot : 9 Students scored 40 or

more marks and it is 9/36 X 100% which is equal to 25%

Stem Leaf

0 9

1 5 6

2 3 4 4 5 5 7 7 8 8 9

3 0 1 1 2 3 4 5 5 6 6 7 8 8 9

4 0 2 3 6 6 7 7 95 0

Central TendencyCentral Tendency refers to the middle value and

mean, median and mode are used to measure it

Which one represent the central tendency depends on the situation

Mean is influenced by extreme values in the data set ( outliers)

Median is not influenced by extreme values in the data set ( outliers)

Mode is referred to the values that occur the most

Investigate when to use Mean, Median or Mode?

The MeanLooking at the middle or center of the data set

and measuring its spread give a better understanding of the data set

The mean of a data set is the statistical for its arithmetic average. It can be found by dividing the sum of the data values by the number of data values

Exercise, Finding MeanThe table below shows the numbers of aces

served by tennis players in their set of the tournament

Determine the mean number of aces for these sets

Number of aces 1 2 3 4 5 6

Frequency 4 11 18 13 7 2

No of Aces Ferquency Product1 4 42 11 223 18 544 13 525 7 355 2 12

Total 55 179

Mean = 179/55 =3.25 aces

Exercise, Finding Mean

The MedianThe median is the middle of an ordered data setThe data set is ordered by listing the data from

smallest to largest. The median split the data in two halves

If there are n data values, the median is:

Exercise for MedianThe following sets of data shows the number of

peas in randomly selected sample of pods. Find the median for each set.

3, 6, 5, 7, 7, 4, 6, 5, 6, 7, 6, 8, 10, 7, 8 ( 15 of them)

The ordered set is:3 4 5 5 6 6 6 6 7 7 7 7 8 8 10n= 15, (n+1)/2 = 8, the median is the 8th data

valueThen the median =6 peas

Exercise for Median Find the median for the following data set: 3,6, 5, 7, 7, 4, 6, 5, 6, 7, 6, 8, 10,7, 8, 9 (16

of them) The ordered data set is: 3 4 5 5 6 6 6 6 7 7 7 7 8 8 9 10 n = 16, (n+1)/2 = 8.5 The median is the average of the 8th and 9th

data values The median is: peas5.6

276

More exercise for Median The data in the table

below shows the number of people on each table at a restaurant, find the median of this data:

Number of people

5 6 7 8 9 10 11 12

Frequency 1 0 3 9 12 7 4 2The total number of data values is the number of tables in the restaurant. It is the sum of the frequencies, which is n=38

5.19239

21

n The median is the average of

the 19th and 20th data values

More exercise for MedianNumber of

people5 6 7 8 9 10 11 12

Frequency 1 0 3 9 12 7 4 2

13 data values of 8 or lessThe 14th to the 25th are all 9s

peoplemedianthe 9299

ModeThe mode is a list of numbers refers to the list of

numbers that occur most frequently.A trick to remember this is that mode starts with

the same two letters that most doesExample: find mode of:9, 3, 3, 44,15,17,17, 44, 15, 15, 27, 40, 8, put in

an order:3,3, 8, 9, 15,15,15, 17, 17, 27, 40, 44, 44The mode is 15. there might be more than one

mode or none

The Spread of DataTo accurately describe a data set, we need not

only a measure of centre, but also a measure of its spread

Commonly used statistics that indicate the spread of a set of data is: The Range

Range is the difference between the maximum or largest data value and the minimum or smallest value

Range = maximum data value - minimum data value

Primary and Secondary Source

Primary Source is the information that is crated at the first stage and it is the original set of information

Secondary Source is information that can be found in different news source and is not original

Example of Scatter PlotWomen Olympic Discus Record: the table shows the result from 1948 to 1996

Year Women (m)1948 41.921952 51.421956 53.691960 55.101964 57.271968 58.281972 66.621976 69.001980 69.961984 65.361988 72.301992 70.061996 69.65

Scatter Plot for the Olympic Discus Result

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Women'sDistance (m)

Women'sDistance (m)

Find a line that best fits the data points

Line of Best Fit (Linear)

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Women'sDistance (m)

Women'sDistance (m)Linear (Women'sDistance (m))

The line of best fit is a straight line that represent all the data on a scatter plot. In this example the relation between x and y is positive or the slope of the best fit line is positive.

Extrapolation of dataOne of the application of the line of the best fit

is to predict the data beyond its available rangeAs an example the women’s Olympic Discus

record for the next Olympic (2016) can be predicted by extending the line of best fit to year 2016

This prediction is called extrapolationCan be predicted to be 85 m

Use the line to predict the future (2016)

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Women's Distance (m)

InterpolationLet assume that the result of the Women’s

Olympics Discus for 1976 was lost and we have the rest

Interpolation can help to find the value close to the missing value

Let’s look at the scatter plot for the Women’s Olympic Discus without the information for 1976

Interpolation

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Women's Distance (m)

Line of best fit (Non-Linear)

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Women'sDistance (m)

Women'sDistance (m)Linear (Women'sDistance (m))Linear (Women'sDistance (m))Polynomial (Women'sDistance (m))

SurveyWhat does a survey mean?When is it conducted? What

does sampling mean?When we have a large amount

of data it is often useful to study only a portion of it to gain insight into the complete set of information

SurveyWhen you sip a spoonful of soup to

test how hot a bowl of soup is, you are sampling. Based on the temperature of the soup in your spoon, you decide if it is too hot to eat

In this case spoon of soup is sample and bowl of soup is the population

SurveyAll members of the population have an equal

chance of being selected.Suppose a survey is conducted to determine the

favorite TV program of students in your schoolOnly students in your class are surveyedSince students in other classes have not been

asked, the sample is not random sampleAll students in your school are not represented

Survey ExamplesExplain why each sample may not provide

accurate information about its population1- A survey of your classmates in used to

estimate the average age of students in your school

2- A survey of senior citizen is used to determine the music that is best liked by Canadian

3- to determine the ratio of domestic cars to foreign cars purchased by Canadians, a person records the numbers of domestic cars and foreign cars in the parking lot of the General Assembly Plant in Oshawa, Ontario

Probability Activity• If you are given 3 coins, the following table shows

the number of possible outcomes you might get As it is shown there are 8 possible equally-likely

outcomesThe probability of each equally-likely outcome is 1/8Disregard which coin has tail or head and put

together those have 2 tails or 2 headsTry to calculate the probability of these possible

outcomesFollow the instruction in the activity and do the

activity

0 T T T1 T T H2 T H T3 T H H4 H T T5 H T H6 H H T7 H H H