UNIT 5: STATISTICS – CHAPTER 14: STATISTICS

Chapter contents

14:02  Sorting data

14:03  Analysing data

14:04  Grouped data

14:05  Dot plots and scatter diagrams

Learning outcomes

Students will be able to:

• Construct, read and interpret graphs, tables, charts and statistical information.
• Collect statistical data using either a census or a sample, and analyse data using measures of location and range.

Explanations

Mathematical terminologies

• Census: a survey of whole population.
• Data: the scores of information collected
• Discrete data: data that is a measure of something restricted to certain limited values.
• Continuous data: data that is a measure of something that can take on any value within reason.
• Dot plot: a graph that uses one axis and a number of dots above the axis.

• Frequency: the number of times an outcome occurs in the data.
• Frequency distribution table: a table that shows all the possible outcome and their frequencies.

• Frequency histogram: a type of column graph showing the outcomes and their frequencies.

• Frequency polygon: a line graph showing outcomes and their frequencies.

• Grouped data: data that is organised into groups and classes.
• Class interval: the size of the groups into which the data is organised.
• Class centre: the middle outcome of a class.
• Mean: the number obtained by ‘evening out’ all the scores until they are equal, the mean is the arithmetic average.
• Median: the middle score for an odd number of scores or the mean of the middle two scores for an even number of scores.
• Median class: in grouped data, the class that contains the median.
• Mode (modal class): the outcome (or class) that occurs most often.
• Outcome: a possible value of the data.
• Outlier: scores that are separated from the main body of scores.
• Range: the difference between the highest and lowest scores.
• Sample: a part, usually a small part, of a large population.
• Random sample: a sample taken so that each member of the population has the same chance of being included.
• Scatter diagram: a graph that uses points on a number plane to show the relationship between two categories

• Statistics: the collection, organisation and interpretation of numerical data.
• Stem-and-leaf plot: a graph that shows the spread of scores without losing the identity of the data.

• Ordered stem-and-leaf plot: the leaves have been placed in order.

• Back-to-back stem-and-leaf plot: this can be used to compare two sets of scores, one set on each side.

Sorting data

The frequency distribution table

When a large amount of data has been collected, it is usually fairly useless until it has been sorted in some way and presented in a form that is easily read. There is a method of sorting data involves using a frequency distribution table. Look at the example below to see how it is done.

Example: Elizabeth surveyed the 30 students in her class to determine the number of children in each family. This resulted in the following data.

If the data is organised in a frequency distribution table, it looks like this:

Note the following features of the frequency distribution table.

Using graphs

Another way of showing the data in an easily read form is by drawing a graph. Two commonly used types of graphs are the frequency histogram and the frequency polygon. The data in the table before can be graphed as a frequency histogram or a frequency polygon.

The frequency histogram

This is a type of column graph where the columns are drawn next to each other. You should note the following features:

• Each axis is labelled.
• The graph has a title.
• The columns are centred on the scores.
• The first column begins one-half of a column width in from the vertical axis.

The frequency polygon

This is a type of line graph. Note that the axes are the same as for the histogram. You should note the following features:

• Each axis is labelled.
• The graph has a title.
• The dots showing the data are joined by straight lines.
• The first and last dot are connected to the horizontal axis as shown.
• The first score is one unit in from the vertical axis.

If a frequency histogram and polygon are both drawn on the same axes, the polygon joins the midpoints of the top of each column in the histogram. This is shown in the diagram below.

.

Let’s do some exercises

1. a. Make a frequency distribution table for the shapes in the rectangle above.

b. How many shapes are there altogether?

Total: 7+6+4+10+7= 34

c. Which shape occurs most frequently?

Star

d. Of which shape are there more: triangles or squares?

Triangles

 

Analysing data

After data has been sorted, certain key numbers can be determined.

Range

The range is defined as the difference or distance between the highest and lowest scores.

Measures of central tendency

The next three key numbers tell us how the scores tend to cluster. They have the names mode, median and mean, and are together called the ‘measures of central tendency’.

Mode

The mode is the outcome that occurs the most often.

Median

After a set of scores has been arranged in order, the median is the ‘middle score’. This is only strictly true if there is an odd number of scores. For an even number of scores, the median is the average of the middle two scores.

Mean

The term ‘average’ usually refers to the mean. To calculate the mean, the sum of the scores is divided by the number of scores.

Finding the mean from a frequency distribution table

To calculate the mean from a frequency distribution table, another column is added: the frequency × outcome column. If x stands for outcome and f stands for frequency, then frequency × outcome can be abbreviated to fx.

Each number in the fx column is obtained by multiplying the outcome and frequency together.

Now the sum of the fx column gives the sum of all the scores and the sum of the frequency column gives the number of scores. So the mean = 49 ÷ 20 = 2·45.

Let’s do some exercises

1. Determine the mean, median, mode and range of each set of scores.

a. 2, 5, 3, 4, 5, 1: mean: (2+5+3+4+5+1)÷5=  4, median: 1, 2, 3, 4, 5, 5: (3+4)÷2= 3.5, mode: 5, range: 5-1= 4

b. 7, 10, 30, 10, 3: mean: (7+10+30+10+3)÷5= 12, median: 3, 7, 10, 10, 30: (10+10)÷2= 10, mode: 10, range: 30-3= 27

 

Grouped data

To draw a frequency distribution table where there are many different numerical outcomes we usually group the outcomes into classes.  The class size should be chosen so that you are using between 5 and 10 classes.

Let’s do some exercises

1. Make a frequency table using the scores below.

51, 40, 63, 81, 73, 73, 35, 49, 55, 61, 31, 60, 44, 54, 87, 56, 97, 50, 37, 53, 60, 58, 84, 77, 63, 80, 60, 53, 42, 38, 52, 43, 50, 72, 80

 

2. Make a table for the scores below.

9, 5, 4, 11, 20, 8, 5, 4, 7, 8, 4, 15, 19, 3, 11, 22, 6, 28, 2, 14, 7, 8, 14, 26, 11, 17, 2, 18, 23, 7, 14, 11, 22, 6, 15, 9, 10, 4, 8, 3

 

Dot plots and scatter diagrams

The dot plot

Features

1. The dot plot uses a number line.
2. By placing one dot plot above another, comparisons can be made easily.
3. It is very similar to a column graph or simple picture graph.

Details

1. The axis should be labelled.
2. Only one axis is used.
3. When drawing the dot plot, decide what part of the number line is to be used.

The scatter diagram

Features

1. The scatter diagram, like the dot plot, is a graph of individual cases. The points representing these cases may form a pattern.
2. The diagram is used to discover relationships between two variables such as Maths Mark and English Mark. The pattern created by the dots tell a story.
3. The graph is not very easy to read, so it is used only by those who know how to use it.

Details

1. The scatter diagram may have a heading.
2. Both axes must be labelled.
3. Each dot represents one person’s scores in two categories. One score is given by the reading on the horizontal axis, and the other by the reading on the vertical axis.
4. Before drawing a scatter diagram, decide on a scale for each axis.

Let’s do some exercises

1. This dot plot shows the scores of students on a driving test.

For the scores shown on the dot plot find:

a. the mode: 19

b. the range: 20-6= 14

c. the median: 17

d. the frequency of the score 17: 3

So that’s all that I can tell you about Chapter 14: statistics. Thank you for reading.

References: Mcseveny, A., Conway, R., Wilkes, S., & Smith, M. (2007). International Mathematics 2 for the Middle Years. In A. Mcseveny, R. Conway, S. Wilkes, & M. Smith, International Mathematics 2 for the Middle Years (pp. 359-395). Pearson Australia

UNIT 4: FURTHER GEOMETRY – CHAPTER 12: CIRCLES

Chapter contents

12:01  Parts of circle

12:02  Circumference of a circle

12:04  Area of circle

Learning outcomes

Students will be able to:

• Classify, construct, and determine the properties of triangles and quadrilaterals.
• Use formulae in calculating perimeter and area of circles.

Explanations

Mathematical terminologies

• Annulus: the area bounded by two circles that have the same centre.
• Arc: part of a circle.
• Chord: an interval joining two points on a circle.
• Circle: the set of all points in a plane that are a fixed distance from a point called the centre (O).
• Circumference: the perimeter of a circle.
• Concentric circles: a group of circles that have the same centre.
• Diameter: an interval that divides the circle in half. It passes through the centre of the circle.
• Pi (π): the ratio of the circumference to the diameter of any circle equal to 3·14 (correct to 2 dec. pl.), 3·1415926 (correct to 7 dec. pl.) or (as a fraction).
• Radius: half of the diameter.
• Sector: the area bounded by two radii and the included arc of a circle.
• Segment: a section bounded by an arc and its chord.
• Semicircle: half of a circle.
• Tangent: a line that touches a circle at only one point.

Parts of circle

In a circle, there are many parts. Here are the parts of circle:

 

Circumference of a circle

The perimeter of circle is called as circumference. Here is the way to find the circumference of a circle:

Let’s do some exercises

1. Calculate the circumference of each circle

a.

Circumference: 3.14×5= 15.7 cm

b.

Circumferece: ×14= 44 cm

 

Area of circle

To find the area of circle, we need to know the radius first. Here is the way to find the area of circle:

Let’s do some exercises

1. Calculate the area of each circle

a.

Area: 3.14×2×2= 12.56 cm²

b.

Area:  ×7×7= 154 cm²

2. Find the radius of each circle, then calculate the area of each circle

a.

Radius: 2÷2= 1 cm

Area: 3.14×1×1= 3.14 cm²

b.

Radius: 7÷2= 3.5 cm

Area: ×3.5×3.5= 38.5 cm²

So that’s all that I can tell you about Chapter 12: circles. Thank you for reading.

References: Mcseveny, A., Conway, R., Wilkes, S., & Smith, M. (2007). International Mathematics 2 for the Middle Years. In A. Mcseveny, R. Conway, S. Wilkes, & M. Smith, International Mathematics 2 for the Middle Years (pp. 314-334). Pearson Australia

UNIT 4: FURTHER GEOMETRY – CHAPTER 11: AREA AND VOLUME

Chapter contents

11:01  Review of area from last year

11:02  Areas of special quadrilaterals

11:03  Formulae for the areas of special quadrilaterals

11:04 Volume of prisms

Learning outcomes

Students will be able to:

• Use formulae and Pythagoras’ Theorem in calculating perimeter and area of circles and figures composed of rectangles and triangles.
• Calculate the surface area of rectangular and triangular prisms and volume of right prisms and cylinders

Explanations

Review of area from last year

Last year we learned how to find the areas of squares, rectangles and triangles. The formulae for these are shown below.

Sometimes a figure needs to be cut into more than one of the basic shapes before its area can be found. Such a figure is called a composite figure. Example:

Let’s do some exercises

1. Calculate the area of each figure.

a.

 Area: ×10×8= 40cm²

b.

Area: (3×9)+(5×4)= 27+20= 47cm²

 

Areas of special quadrilaterals

If we know how to find the areas of rectangles and triangles, we can calculate the areas of many other plane shapes.

Let’s do some exercises

1. Calculate the area of each figure.

a.

Area: 9×7= 63 cm²

b.

Area: ×12×(8+12)= ×12×20= 120 cm²

 

Formulae for the areas of special quadrilaterals

These are the formula of quadrilaterals:

Let’s do some excercises

1. Calculate the area of each figure.

a.

Area: 12×8= 96 cm²

b.

Area: ×15×12= 90 cm²

 

Volume of prisms

A prism has a special pair of parallel faces called bases. If we cut the prism parallel to its base, the cross-section formed is always the same. The number of cubic units in a prism is equal to the number of units in the base times the number of layers.

Let’s do some exercises

1.  Calculate the volume of each prism.

a.

Area: 3×7×2= 42 cm²

b.

Area: ×6×3×6= 54 cm²

 

So that’s all that I can tell you about Chapter 11: area and volume. Thank you for reading.

References: Mcseveny, A., Conway, R., Wilkes, S., & Smith, M. (2007). International Mathematics 2 for the Middle Years. In A. Mcseveny, R. Conway, S. Wilkes, & M. Smith, International Mathematics 2 for the Middle Years (pp. 283-302). Pearson Australia