Exercise 2

Measures of Dispersion

1) The math test scores of five students are : 92, 88, 80, 68 and 52.

2) x = {14, 6, 3, 2, 4, 15, 11, 8, 1, 7, 2, 1, 3, 4, 10, 22, 20}

(a) Find the mean
(b) Find the median
(c) Find the interquartile range: Q1 & Q3
(d) Find the semi-interquartile range
(e) Construct a box and whisker for the data set


Probability

1) The probability the student may pass mathematics is 2/3 and the probability he will pass physics is 3/4. If the probability that he will pass both subjects is 1/2, what is the probability that he will pass at least one subject?

2) A vacuum cleaner salesmen MJ must make two calls per day, one in the morning and one in the afternoon. MJ has probability of 0.4 of selling a cleaner on any call. The morning and afternoon results are independent of each other. Find the probability that, in one day:

a) MJ sells just one cleaner
b) MJ doesn't sell any cleaner

Permutation & Combination

1) In the Match of the Day's goal of the month competition, you had to pick the top 3 goals out of 10. How many ways can you choose the top 3 goals?

2) In class of 20 students, a first and second prize are to be awarded. In how many different ways can this be done?

Exercise 1

Set Theory

1) There are 50 people on tour. One day, 26 people went on the morning cruise and 29 to the evening barbecue. Using Venn diagrams, or otherwise, answer the following questions.

(a) It was thought that 4 people went to both events and 1 person to neither. Explain why this was not possible. 

(b) Find the least number and the greatest number of people who could have gone to both events.

2) Mary has 50 counters. Some of the counters are square, the remainder are round. There are 11 square counters that are green. There are 15 square counters that are not green. Of the round counters, the number that are not green is double the number that are green.

By drawing Venn diagram, or otherwise, find the number of counters that are

(a) round,              (b) round and green,               (c) not green.

Statistical Data

1) For each of the data set given below, identify whether it is discrete or continuous.

    (a) The number of students in Cosmopolitan college in the year of 2013
    (b) The time taken for a laptop to shut down

2) For each of the following, state whether it is a primary or a secondary data.

    (a) A student recorded the time taken by a batch of desktops to shut down.
    (b) Jamilah is the owner of a book store. She is looking onto the account prepared by her clerks to find out the amount of profit she made in last month.

3) From the following information, distinguish whether it is quantitative or qualitative.

    (a) The height of each football player
    (b) The colour of car passing by maria's house

Statistical Representation

1) Cameron observed the make of 100 cars passing his house. Draw a pie chart to show him results. Calculate the fraction of the whole that each sector represents.

2) (a) An article in a newspaper reported that the number of crimes had been reduced by half from 1991 to 2001. The article contained the bar chart shown here. Explain why this bar chart might considered misleading.  

(b) The histogram alongside shows distribution of times taken by a group of students to travel to school. 11 students took at least 5 but less than 10 minutes. Complete the table in the answer space.

Measures of Central Tendency

1) A marksman, firing at a target, can score from 0 to 6 points. After firing 15 shots, his score were as follows:

Find (a) the mean (b) the mode and (c) the median.

2) The number of ice-creams sold in s shop each month is shown in the table.

(a) Find the range (b) Calculate the mean (c) Find the median

Topic 7: Permutation and Combination

Permutation - The number of different ways that a certain number of objects can be arranged in order from a huge number of objects.

• An ordered list, order matters.
• Keyword(s) : Arrangement

Formula: 

Combination - The number of different ways that a certain number of objects as a group can be selected from a large number of objects.

• Un - ordered group/list
• Keyword(s) : choice, selection, election.

Formula : 

Example 1,

How many ways can you arrange these 4 objects.

Permutation :

N! / (N - n)!  = 4! / (4 - 4)! = 4 x 3 x 2 x 1 / (1)  = 24 ways

Combination :

N! / n!(N - n)! = 4! / 4!(4 - 4)! = 4 x 3 x 2 x 1 / 4 x 3 x 2 x 1 x 1 = 1 ways


Example 2,

How many ways can you arrange of 7 objects picking 5 at a time?

Permutation:

N! / (N - n)! = 7! / (7 - 5)! = 7 x 6 x 5 x 4 x 3 x 2 x 1 / 2 x 1 = 2520 ways

Combination:

N! / n!(N - n)! = 7 / 5!(7-5)! = 7 x 6 x 5 x 4 x 3 x 2 x 1 / 5 x 4 x 3 x 2 x 1 x 2 x 1 = 21 ways

Topic 6: Probability

Define:
Probability is the chance that something will happen - how likely it is that some event will happen.

For a sample space S with a finite number of equally likely outcomes.

Probability of an event E,


Where n (E) is the number of possible outcomes in the event E, n (S) is the number of possible outcomes in the sample space S.



Example 1,

There are 10 balls in a bag. One is Red, two are Blue, three are Yellow and four are Green. A ball is taken out without looking. What is the probability that it is.

Example 2,

The probability that a football team will win their next match is 1/4. What is the probability that the team will not win?




Tree Diagram

Independent events and their probabilities can be shown on a tree diagram. Each event is represented by a branch.

Example 1,

A coin is flipped twice. Draw a thee diagram to show all the possible outcomes.

Outcomes Probabilities
 HH 1/2 x 1/2 = 1/4






HT 1/2 x 1/2 = 1/4
TH 1/2 x 1/2 = 1/4






TT 1/2 x 1/2 = 1/4

Topic 5: Measures of Dispersion

Objectives

Find and calculate the different types of spreads namely Standard Deviation/Variance of ungrouped and grouped data, The Range and the Quartiles.

Use the different types of measure of spreads to compare different sets of data.

Explain the relationship between variance and standard deviation.

Range

The range is the difference between the largest and smallest values in the data set. i.e.

Range = Xmax - Xmin

Example 1,

Find the range of values 1, 4, 5, 6, 9, 10, 15

solution:

Range = Xmax - Xmin

           = 15 -1

           = 14

Quartiles and Interquartile Range

The lower quartile, Q1, is the value 25% of the way through the distribution and the upper quartile, Q3, is the value 75% of the way through the distribution.

Example 2,

Find the median and interquartile range of the following numbers:

2, 3, 3, 9, 6, 6, 12, 11, 8, 2, 3, 5, 7, 5, 4, 4, 5, 12

Solution: 

Example 3,

Find the median and interquartile range of the following distributions:

Q1 = 52  Q2 = 59  Q3 = 70  max = 84 and min = 46

Standard Deviation and Variance

Standard Deviation 

Perhaps the most commonly used measure of dispersion is the standard deviation which is the square root of the variance.

Standard deviation of a set of values can be obtained by using the formula below.

Variance 

Example 1,

Find the variance and standard deviation of values $2, $3, $5, $7 and $10.

Topic 4: Measures of Central Tendency

Objectives

Use the three measures of central tendency (mean, median and mode) in comparing and contrasting sets of ungrouped data. Select and apply a suitable measure given the necessary information

Measures of central tendency are single values that are typical and representative for a group of numbers. They are also called measures of locations. A representative values of location for a group of numbers is neither the biggest nor the smallest but is a number whose values is somewhere in the middle of the group. Such measures are often used to summarize a data set and to compare one data set with another.

Mean

The average value of set of data.

Appropriate for describing measurement data. e.g. heights of people, marks of student papers.

Often influenced by extreme values.

Median

The median of a set of values is defined as the value of the middle item when the values are arranged in ascending or descending order of magnitude.

Mode

The most frequent or repeated value.

In the case of a continuous variable, it is possible that no two values will repeat themselves. In such a situation, the mode is defined as the point of highest Frequency Density, i.e., where occurrences cluster most closely together.

Like the median, the mode has very limited practical use and cannot be subjected to arithmetical manipulation.

However, being the value that occurs most often, it provides a good representation of the data set.

Example 1,

The number of goals in a series of 12 matches are 3, 2, 4, 3, 2, 6, 3, 2, 2, 4, 1, 4. 

Find a) the mean b) the mode and c) the median number of goals.

1, 2, 2, 2, 2, 3, 3, 3, 4, 4, 4, 6

a) the mean - 36 / 12 = 3

b) the mode - 2

c) the median - 3 + 3 / 2 = 3

Example 2,

A gardener planted some cucumber seeds in each of 11 pots. The number of seeds that germinated in each pot were 2, 7, 2, 1, 1, 5, 3, 1, 7, 1, 3.

Find a) the mean b) the mode and c) the median

1, 1, 1, 1, 2, 2, 3, 3, 5, 7, 7

a) the mean - 33 / 11 = 3

b) the mode - 1

c) the median - 2 + 2 / 2 = 2

Topic 3: Statistical Representation

Histogram 

The most appropriate method of illustrating a grouped frequency distribution is by means of a histogram.

Unlike bar charts, there are no gaps between the bars of a histogram. The bars are drawn between the class boundaries, and the area of the bar is proportional to the frequency.

It is constructed by first making off class boundaries on the x-axis and then drawing rectangles whose heights equal the class frequencies if classes are of equal width.

HISTOGRAM WITH EQUAL CLASS WIDTH

Example 1,

The diameters of 30 pipelines were measured and the information grouped as shown. Draw a histogram to illustrate the data.

Solution:

HISTOGRAM WITH UNEQUAL CLASS WIDTH

If the classes are of unequal width, then we may use Frequency Density.

   

Example 2, 

The diameters of 34 pipelines were measured and the information grouped as shown. Draw a histogram to illustrate the data.

Pie Chart

A pie chart is a circular diagram split into sections

The angle of each sector of the pie hart is in proportion to the amount of information it represents

The angles of the sectors in a pie chart add up to 360°

To draw a pie chart:

1. Draw a circle

2. Use a protractor to measure the angle for each sector

3. Either label the sectors or use a key.

Example 1, 

A travel brochure contains 24 picture from different countries. The table shows how many pictures there are from each country.

Bar Chart

A Bar Graph (also called Bar Chart) is a graphical display of data using bars of different height.

Bar Graph are good when your data is in categories such as "Comedy", "Drama" and etc. But when you have continuous data such as a person's height then use a histogram.

It is best to leave gaps between the bars of a Bar Graph, so it doesn't look like a Histogram.

Example 1, 

The colours of 30 cars in a park are shown in the frequency table.

a) Complete the bar chart to represent this information.

Topic 2: Statistical Data

Quantitative and Qualitative

Numerical data is quantitative
- e.g. cost of a shirt

Non-numerical data is qualitative
- e.g. the colour of a shirt

Example 1,

Qualitative:

- blue/grey color

- texture shows brush strokes of oil paint

- peaceful scene of the country

Quantitative:

- weighs 9 pounds 

- cost $500

- width & height 13.33 x 8.33"

Example 2,

Qualitative: 

- frothy appearance

- white cup

- strong taste

Quantitative:

- serving up 5 inches in height 

- 12 ounces of latte

- cost $4.50

Discrete and Continuous

Discrete data can be counted. They can take particular values.
- e.g. number of children, number of trees in a garden

Continuous data results when measuring things like length, time and mass. It cannot be measured exactly.
- e.g the time taken to run 100m. It could be 9s or 9.8s or 9.81s. It can also be measured more accurately

Example 1,

a) The number of people sleeping in Biology class

Answer: Discrete

b) The shoe sizes of the British women's hockey team

Answer: Continuous

Example 2,

a) The height of the Chinese basketball team

Answer: Continuous

b) The number of languages of a person speaks

Answer: Discrete

Primary and Secondary data

Primary Data

Data that is collected for a specific purpose. Basically data collected in raw form directly from a source for a specific purpose. e.g. jotting down important points during a lecture, interviewing and etc.

Secondary Data

Data collected from sources that have already collected it. It is second hand or re-used data that is collected from primary data. e.g. data collection from books and newspaper.

Example 1,

a) Data collected by a student for his/her thesis or research project.

Answer: Primary data

b) Census data being used to analyze the impact of education on career choice and earning.

Answer: Secondary data

Topic 1.2: Example of Set Theory

Example 1,

x = {2, 3, 7, 15, 18}

y = {7, 9, 11, 13, 14}

x n y = 7

x u y = 2, 3, 7, 9, 11, 13, 14, 15, 18

u = {7, 8, 9, 10, 11, 12, 13}


Example 2,

A = {8, 10, 12}

B = {10, 12, 13}

C = {7, 8}

A u B = {8,10, 12, 13}

A n B = {10, 12}

Topic 1: Set Theory

A set is a collection of objects such as numbers, points, shapes, ideas, etc.

The individual objects is a set are called the members or elements of the set.

There are 3 ways of writing down sets:

i) by description or by using the set builder notation.

ii) by listing its element  

  

   e.g. C = {a, b, c, d}

iii) by using a Venn diagram

Finite set

A finite set is a set which has finite number of members.

e.g. A = {2,4,6,8, ... 100}

Infinite set

An infinite set is a set which is not finite 

e.g [all triangles]

Universal set

A universal set is the set which contains all the available elements, denoted by E.

Empty set

The empty set or the null set is the set having no elements. It is denoted by [ ] or  ф

Note:  [ 0 ] and [ ф ] are not empty set as they have the element [ ] and ф respectively.

Subset

If every element of a set B is also a member of a set A, then we say B is a subset of A and write B ⊂ A. 

Intersection

The intersection of two sets A and B is the set of elements which is common to both A and B. it is denoted by A ∩ B and is read A intersect B. the shaded portion of the figure below shows A ∩ B.

Union

The union of two sets A and B is the set of elements which is in A or in B or in both A and B. it is denoted by A ∪ B and is read 'A union 'B. The shaded portion of the figure below shows A ∪ B.

Sets: Union and Intersection (Basic Video)