# FORM THREE MATHEMATICS STUDY NOTES TOPIC 3-4.

**TOPIC 3: STATISTICS**

**Mean**

Calculating the Mean from a Set of Data, Frequency Distribution Tables and Histogram

Calculate the mean from a set of data, frequency distribution tables and histogram

Measures of central tendency:

The arithmetic mean

Example 1

The masses of some parcels are 5kg, 8kg, 20kg and 15kg. Find the mean mass of the parcels.

**Solution**

Total mass = (5 + 8 + 20 + 15) kg = 48kg

The number of parcels = 4

The mean mass = 48kg ÷ 4 = 12kg

The arithmetic mean used as measure of central tendency can be misleading as can be seen in the following example.

Example 2

John
and Mussa played for the local cricket team. In the last six batting
innings, they scored the following number of runs.John: 64, 0, 1, 2, 4,
1;Mussa: 15, 20, 13, 11 , 10, 3.Find the mean score of each player.
Which player would you rather have in your team? Give a reason.

**Solution**

John’s mean = (64 + 0 + 1 + 2 + 4 + 1) ÷ 6 = 12

Mussa’s mean = (15 + 20 + 13 + 11 + 10 + 3) ÷ 6 = 12

Each
player has the same mean score. However, observing the individual
scores suggests that they are different types of player. If you are
looking for a steady reliable player, you would probably choose Mussa.

Often it is possible to use the mean of one set of numbers to find the mean of another set of related numbers.

Suppose a number a is added to or subtracted from all the data. Then a is added to or subtracted from the mean.

Suppose
the n values are 𝑥!+𝑥! + 𝑥! .........+𝑥!. Multiply each by a, and
we obtain 𝑎𝑥!+𝑎𝑥! + 𝑎𝑥! .........+𝑎𝑥!. So we see that the mean
has been multiplied by a.

Interpreting the Mean Obtained from a Set Data, Frequency Distribution Tables and Histogram

Interpret the mean obtained from a set data, frequency distribution tables and histogram

Measures of central tendency from frequency tables

If the data has already been put into a frequency table, the calculation of the measures of central tendency is slightly easier.

Exercise 1

Juma rolled a six- sided die 50 times. The scores he obtained are summarized in the following table.Calculate the mean score

Score (x) | 1 | 2 | 3 | 4 | 5 | 6 |

Frequency (f) | 8 | 10 | 7 | 5 | 12 | 8 |

**Solution**

10 scores of 2 give a total 10 x 2 = 20

8 scores of 1 gives a total 8 x 1 = 8

And so on, giving a total score of

8 x 1 +10 x 2+7 x 3 + 5 x 4 + 12 x 5 + 8 x 6 = 177

The total frequency = 8 + 10 + 7 + 5 + 12 + 8 = 50

The mean score = 177 ÷ 50 = 3.34

**Medium**

The Concept of Median

Explain the concept of median

Mr.Samwel
owns a small factory. He earns about 4,000,000/- from it each year. He
employs 4 people. They earn 550,000/-, 500,000/-, 450,000/- and 400,
000/-.The mean income of these five people is(4,000,000 + 550,000 +
500,000 + 450,000 + 400,000 ÷ 5 = 1,180,000/-

If
you said to one the employees that they earned about 1,180,000/- each
year they would disagree with you. In this type of situation when one of
the values is different from the others (as in Example 2), the mean is
not the best measure of central tendency to use.Arrange the incomes in
increasing order of size as follows:

The
value that appears in the middle is called the median. In this case the
value of 500,000/- is a much better idea of the average wage earned by
the employees. The median is not affected by isolated values (sometimes
called rogue values) that are much larger or smaller than the rest of
the data.

If the data consists of an even number of values, find the mean of two middle values as shown in the next example.

The Medium from a Set of Data

Calculate the medium from a set of data

Example 3

Find the median of the numbers: 12, 23, 10, 8, 22, 14, 30, and 18.

**Solution**

Arranging in increasing order of size, we get 8 10 12 14 18 22 23 30

Median = (14 + 18) ÷ = 16

The Median using Frequency Distribution Tables and Cumulative Curve

Find the median using frequency distribution tables and cumulative curve

Example 4

Juma rolled a six- sided die 50 times. The scores he obtained are summarized in the following table. Calculate the modianl score

**Solution**

here
are 50 items of data, so if you arrange them in order of size, the
positions are1 .................... 25 and 26 ................. 50. The
median will be the average of the 25th and 26th number.

In
the table there are 8 scores of 1, followed by 10 scores of 2. This
gives you 8 + 10 = 18 numbers. These are then followed by 7 scores of 3.
This gives 18 + 7 = 25 numbers. It follows that the 25th number is a 3.
The 26th number must be the first number in the next group, which is a
4.

The median is then = (3 + 4) ÷ 2 = 3.5

The Median Obtained from the Data

Interpret the median obtained from the data

Exercise 2

- The times of five athletes in the 100 m were: 12.5 s, 12.9s, 14.8s, 15.0s, 25.2s. Find the median time. Why is the median a better measure of central tendency to use than the mean?
- Iddi has 6 maths tests during a school term. His marks are recorded below. Find the mean and the median mark. Explain why the median is a better measure of central tendency than the mean 73 78 82 0 75 86
- The table below gives the percentage prevalence of HIV infection in female blood donors for the years 1992 to 2003. Find the mean and median of these figures.

1992 | 1993 | 1994 | 1995 | 1996 | 1997 | 1998 | 1999 | 2000 | 2001 | 2002 | 2003 |

5.9 | 6.2 | 4.8 | 9.4 | 8.2 | 11.6 | 11.8 | 12.6 | 13.3 | 13.7 | 12.3 | 11.9 |

**Mode**

The Concept of Mode

Explain the concept of mode

The
mode is value that occurs most often in a set of data.This is another
measure of central tendency. It is possible for data to have more than
one mode.

Data with two modes are said to be bi – modal.Why mode? The mode is often important to know. For example:

- If you ran a shoe shop you would want to know the most popular size.
- If you ran a restaurant you would want to know what type of food is ordered most.

The Mode

Calculate the mode

Example 5

State the mode for the following sets of numbers:

- 0, 0, 1, 1, 1, 2, 2, 3, 4, 5, 5
- 58, 57, 60, 59, 50, 56, 62
- 5, 10, 10, 10, 15, 15, 20, 20, 20, 25

**Solution**

- 1 occurs most (3 times): The mode is 1
- All the numbers appear once: There is no mode.
- There are three 10s and three 20s: Modes are 10 and 20.

Exercise 3

- Ten pupils were asked how many brothers or sisters they had. The results are recorded below. Find the mode number 0, 1 , 1, 2, 5, 0, 1 3 , 1 and 4.
- Eight motorists were asked how many times they had taken the driving test before they passed. The results are recorded below. Find the mode number. 14213141
- Give examples of where the mode is a better measure of central tendency than either the mean or the median.
- Find the mode of these sets of numbers.

- 0, 1, 1, 3, 4, 5, 5, 5, 6, 7, 8
- 3, 8, 4, 3, 8, 4, 3, 8, 8, 3, 3, 4
- 5, 12, 6, 5, 11, 12, 5, 5, 8, 12, 7, 12
- 3, 6, 2, 8, 2, 1, 9, 12, 15

Finding the Mode using Frequency Distribution and a Histogram

Find the mode using frequency distribution and a histogram

Grouped data

Suppose
a set of data consists of many different values, such as heights of
people measured to the nearest centimeter. Then the data is grouped, for
example into 160 – 165 cm, and so on. If the data has been grouped
together in classes, then unless you have a list of all the individual
values, you only know approximately what each value is. For this reason,
you can only estimate the mean and the median. Also, if all the values
are different, you do not have a single value as the mode. Instead you
have a modal class, as shown in the example below.

Data
grouped in classes can be illustrated by a histogram.Suppose one of the
intervals is from 10 to 19, where data has been rounded to the nearest
whole number. The class limits are 10 and 19. The data in this interval
could be as low as 9.5 or as high as 19.5. These are the class
boundaries. The width of the interval is the difference between the
class boundaries, in this case it is 10.

The
histogram consists of rectangles between the class boundaries, with
height corresponding to the frequency. The area of each rectangle is
proportional to the frequency.

Example 6

The examination results (rounded to the nearest whole number %) are given for a group of students.

Mark (%) | 30 – 39 | 40 -49 | 50 – 59 | 60 - 69 | 70 - 79 |

Frequency | 5 | 3 | 20 | 2 | 10 |

- Draw a histogram
- state the modal class

**Solution**

For
a histogram, the horizontal axis is for the data values, and the
vertical axis is for the frequencies. So label the horizontal axis with
the marks from 30 to 80. To indicate that the axis does not start at 0
put a zig – zag to the left of 30. Label the vertical axis with
frequencies from 0 to 20. The first interval has limits 30 and 39. The
class boundaries are 29.5 and 39.5. It has a frequency of 5. So draw a
box covering the interval, and with height 5. Repeat with the other
intervals

Interpreting the Mode Obtained from the Data

Interpret the mode obtained from the data

Example 7

The examination results (rounded to the nearest whole number %) are given for a group of students.

Mark (%) | 30 – 39 | 40 -49 | 50 – 59 | 60 - 69 | 70 - 79 |

Frequency | 5 | 3 | 20 | 2 | 10 |

Estimate the mode

**Solution**

To estimate the mode, there are two methods.

**By drawing:**Use the histogram of the first part.Then proceed as follow;

- Step 1: Draw a straight line from the top left hand corner of the rectangle of the modal class, to the top left hand corner of the rectangle of the class to the right of the modal class.
- Step 2: Draw a line from the top right hand corner of the rectangle of the modal class,to the top right of the modal class to the left of the modal class.
- Step 3: Find where these two lines intersect. This gives the mode as 54 on the horizontal axis.

**By calculation:**Let

- fM = frequency of the modal group
- fR = frequency of the group to the right of the modal group
- fL = frequency of the group to the left of the modal group
- W = width of the modal group
- L = lower class boundary of the modal group

**TOPIC 4: RATES AND VARIATIONS**

**Rates**

A rate is found by dividing one quantity by another.

Rates of Quantities of Different Kinds

Relate rates of quantities of different kinds

For
example a rate of pay consists of the money paid divided by the time
worked. If a man receives 1,000 shilling for two hours work, his rate of
pay 1000 ÷ 2 = 500 shillings per hour. From the above example, we find
out that

Example 1

1. A man is paid 6,000/= for 8 hours work.

- What is his rate of pay?
- At this rate, how much would he receive for 20 hours work?
- At this rate, how long must he work to receive 30,000 shillings?

*Solution*:
Quantities of the Same Kind

Relate quantities of the same kind

Example 2

A
student is growing plants she measures the rate at which two of them
are growing. Plant A grew 5cm in 10 days, and plant B grew 8cm in 12
days. Which plant is growing more quickly?

Exercise 1

1. A woman is paid 12,000/= for 8 hours work.

Converting Tanzanian Currency into other Currencies

Convert Tanzanian currency into other currencies

Different
countries have different currencies. Normally money is changed from one
currency to another using what is called a Rate of Exchange.

This makes trade and travel between countries convenient.

Conversion of money is done by multiplying or dividing by the rate of exchange.

Eg.
If at a certain time there are 1,100 shillings to each UK pound (£), to
go from £to shillings, multiply by 1,100, and to go from shillings to
£divide by 1,100.

**NB:**The rate of exchange between two countries varies from time to time.

Example 3

Suppose the current rate of change between the Tanzanian shillings and the Euro is 650 Tsh per Euro.

- A tourist changes 200 euros to Tsh. How much does he get?
- A business woman changes 2,080,000 Tsh to euros. How much does she get?

Exercise 2

At a certain time there are 600 Tsh to one US dollar ($).

**Variations**

The Concept of Direct Variation

Explain the concept of direct variation

Some
quantities are connected in such a way that they increase and decrease
together at the same rate. Afar example if one quantity is doubled the
other quantity is also doubled. These quantities are Directly
Proportional or Vary Directly.

Eg. If a car is driven at a constant speed, the distance it goes is directly proportional to the time taken.

Also the amount of maize you buy is directly proportional to the amount of money you spend.

Problems on Direct Variations

Solve problems on direct variations

Example 4

1. Suppose different weights are hung from a wire. The extension of the wire is proportional to the weight hanging.

Suppose a weight of 2kg gives an extension of 5cm.

Find an equation giving the extension e cm in terms of weight w kg. Find the weight for an extension of 3cm.

Example 5

Given
that y is proportional to x such that, when x = 40, y = 5. Find an
equation giving y in terms of x and use it to find (a) y when x = 15 (b)
x when y = 20.

Exercise 3

The variables m and n are directly proportional to each other such that when m = 3, n= 12.

Graphs of Direct Variation

Draw graphs of direct variation

Example 6

The linear equation graph at the right shows that as the

*x*value increases, so does the*y*value increase for the coordinates that lie on this line.
This is a graph of

**direct variation**
The Concept of Inverse Variation

Explain the concept of inverse variation

In
some cases one quantity increase at the same rate as another decrease.
For example, if the first quantity is doubled, the second quantity is
halved.

In
this case the quantities vary inversely, or they are inversely
proportional. e.g. The number of men employed to dig a field is
inversely proportional to the time it takes. Also the time to travel a
journey is inversely proportional to the speed. We use the same symbol
(∝) for proportionality.

Problems on Inverse Variations

Solve problems on inverse variations

Example 7

1. Suppose a mass of a gas is kept at a constant temperature. The volume of the gas is inversely proportional to its pressure.

^{3}when the pressure when the is 250kg/m

^{3}, find the formula giving the volume vm

^{3}in terms of the pressure P kg/m

^{2}. What is the volume when the pressure is increased to 1,000kg/m

^{2}?

Example 8

Given
that y is inversely proportional to x, such that x = 8 when y = 15.
Find the formula connecting x and y by expressing y in terms of x and
use it to find (a) y when x = 10, (b) x when y = 3

Exercise 4

The quantities p and q are inversely proportional to each other such that when q = 20, p=1.2

- Find the equation giving p in terms of q
- Find q when p = 0.5
- Find p when q = 160

Given
that y is inversely proportional to x such that when y = 6, x =7. Find
the equation connecting x and y by expressing x in terms of y and hence
find x when y = 36

The
number of workers needed to repair a road is inversely proportional to
the time taken. If 12 workers can finish the repair in 10 days, how long
will 30 workers take?

Graphs Relating Inverse Variations

Draw graphs relating inverse variations

The graph of y against x is shown for which y = 3 when x =2

**Proportion to powers:**

Sometimes
a quantity is proportional to a power of another quantity. For example
the area A of a circle is proportional to the square of its radius r,

So A ∝r

^{2}or A= kr^{2}
Example 9

1.
The mass of spheres of a certain metal is proportional to the cube of
their radii. A sphere of radius 10cm has mass 42kg. Find the formula
giving the mass m kg in terms of radius r cm. Find the radius of the
sphere with mass 5.25 kg.

Example 10

Given
that M is proportional to the square of N and when N = 0.3, M = 2.7.
Find the equation giving M in terms of N, and hence find the value of:

- M when N = 1.5
- N when M = 0.3

Joint Variation in Solving Problems

Use joint variation in solving problems

If a quantity varies as the product of two other quantities then it varies jointly with them. eg. If y = 3vu

^{2}, then y varies jointly with v and u^{2}.
Example 11

1. Suppose a mass of a gas with volume Vm

^{3}is under pressure P kg/m^{2}and has absolute temperature T^{0}.^{2}, the volume is 0.5m

^{3}. Find the formula for the volume in terms of T and P.

Example 12

m
varies jointly with p and q such that when p = 12 and q = 5 then m= 15.
Find m in terms of p and q and hence find m when P = 3 and q = 28

Exercise 5

1. M is inversely proportional to the cube of N, when N =2 then M = 20.

- Find an equation giving M in terms of N.
- Find M when N = 4
- Find N when M = 5.

2. P is inversely proportional to the square root of Q. When Q = 16 then P =5.

- Find an equation connecting P and Q expressing P in terms of Q.
- Find P when Q = 9

3.
When a body is moving rapidly through the air, the air resistance R
newtons is proportional to the square of the velocity Vm/s, At a
velocity of 50m/s, the air resistance is 20N.

- Find R in terms of V
- Find the resistance at 100m/s.

4. B varies jointly with A and the inverse of C. When A = 3 and C = 12 then B = 20.

- Find B in terms of A and C.
- Find B when A = 8 and C= 2

5.
The mass m kg of a solid wooden cylinder varies with the height h (m)
and with the square of the radius r (m). If v = 0.2 and h = 1.4, then M =
150. Find m in terms of h and r.

Joint variation leading to areas and volumes

Many formulas for areas and volumes involve joint variation. For example the volume of a cylinder is given by v = πr

^{2}h.
So the volume varies jointly with the height and the square of the radius. i.e v∝r

^{2}h.
Example 13

1.
A cylinder has radius 3cm and volume 10cm3. If the radius of the base
is increased to 4cm without altering the height of the cylinder what
effect does this have on the volume?

Example 14

A
pyramid has a square base. If the height decreases by 10% but the
volume remains constant, what must the side of the base increase by?
(i.e What increase in the side will of set the decrease in the height?).

Exercise 6

1. A box has a square base of side 5cm. The volume of the box is 56cm

^{3}. If the sides increase by 10%, without the height changing, what is the new volume of the box?
2. A cone has volume 30cm

^{3}. If the radius increases by 10% and the height by 5%, what is the new volume of the cone?
3.
A water tank holds 1,000 liters, and is in the shape of cuboids. The
lengths of the sides of the base are enlarged by a scale factor of 1.4
without altering the height. What volume will the tank now hold?

4. The height of a cylinder is reduced by 20%. What percentage change is needed in the radius, if the volume remains constant?

**MATHEMATICS FORM THREE OTHER**

**TOPICS**

**FORM THREE MATHEMATICS STUDY**

**NOTES TOPIC 1-2.**

**FORM THREE MATHEMATICS STUDY NOTES TOPIC 3-4.**

**FORM THREE MATHEMATICS STUDY NOTES TOPIC 5.**

**FORM THREE MATHEMTICS STUDY NOTES TOPIC 6-7.**

**FORM THREE MATHEMTICS STUDY NOTES TOPIC 8.**

**O'LEVEL MATHEMATICS NOTES**

**FORM ONE MATHEMATICS STUDY NOTES****FORM TWO MATHEMATICS STUDY NOTES****FORM THREE MATHEMATICS STUDY NOTES****FORM FOUR MATHEMATICS STUDY NOTES**

## No comments