Measures of average in grouped and continuous data

The mean

We already know how to find the mean from a frequency table. Finding the mean for grouped or continuous data is very similar.

The grouped frequency table shows the number of CDs bought by a class of children in the past year.

 

Number of CDs	Frequency (f)
0-4	10
5-9	12
10-14	6
15-19	2
>19	0

• We know that 10 children have bought either 0, 1, 2, 3 or 4 CDs, but we do not know exactly how many each child bought.
• If we assumed that each child bought 4 CDs, it is likely that our estimate of the mean would be too big.
• If we assumed that each child bought 0 CDs, it is likely that our estimate would be too small.
• It therefore seems sensible to use the mid-point of the group and assume that each child bought 2.

Finding the mid-points of the other groups, we get:

 

Number of CDs	f	Mid-point, x	fx
0-4	10	2	20
5-9	12	7	84
10-14	6	12	72
15-19	2	17	34
>19	0	-	0

The mean is 

Remember: This is only an estimate of the mean.

The median

As explained previously, the median is the middle value when the values are arranged in order of size.

As the data has been grouped, we cannot find an exact value for the median, but we can find the class which contains the median.

 

Number of CDs	Frequency (f)
0-4	10
5-9	12
10-14	6
15-19	2
>19	0

There are 30 children, so we are looking for the class which contains the (30 + 1) ÷ 2 = 15¹₂th value. The median is therefore within the 5-9 class.

The mode

The mode is the most common value.

We cannot find an exact value for the mode, and therefore give the modal class. The modal class is 5-9.

Direct and indirect proportion - Higher

We sometimes need to find an equation for two quantities that are in proportion.

If a and b are in direct proportion then we can write this as a ∝ b.

∝ is the symbol for proportionality

We can use the example from the previous page.

It costs 72p for 12 pencils.

Work out an equation connecting cost and number of pencils, then find out how much 30 pencils will cost.

Let a be the cost in pence and b the number of pencils.

a ∝ b

Which means:

• a = k x b
• 72 = k x 12
• k = 72 ÷ 12
• k = 6

We can find the equation is a = 6b.

Now, we can calculate the price for 30 by substituting b with 30.

So:

• a = 6 x 30
• a = 180p

If a and b are in indirect proportion then a ∝ ¹/_b.

We can work out the following questions in the same way.

Example

The time taken to dig a hole is indirectly proportional to the number of people doing the digging.

It takes 4 people 6 hours to dig the hole.

Find an equation connecting the time, t, to the number of people digging, d.

Solution

t and d are indirectly proportional, so:

• t ∝ ¹/_d

Which means:

• t = k x ¹/_d
• 6 = k x ¹/₄

The equation is t = ²⁴/_d

Question

How long would it take 8 people to dig the hole?

Transformations and enlargements

If we translate an object, we move it up or down or from side to side. But we do not change its shape, size or direction.

QQuestion

Which of the following triangles P, Q or R is a translation of triangle ABC?

AReveal answer

Remember

When we translate an object, every vertex (corner) must be moved in the same way.

Example:

Triangle PQR has been translated 3 squares down and 4 squares to the right. All of the vertices have been translated in the same way, and the object and its image are exactly the same shape and size.

Calculating with upper and lower bounds - Higher

In these calculations we need to think about:

• whether we need upper or lower bounds
• whether to add, subtract, multiply or divide
• how sensible our answers are

Question

A sack of sand weighs 20 kg measured to the nearest kg. It is used to fill bags that will contain 250 g of sand measured to the nearest 10 g. Work out the maximum number of bags that can be filled.

Solution

To find the maximum number of bags that can be filled, we need the maximum possible total weight of the sack and the minimum possible weight in each bag. Then we need to divide the maximum sack weight by the minimum bag weight.

The maximum weight for the sack is 20.5 kg.

The minimum weight for each bag is 245 g.

Before we can divide, the units need to be the same. Convert kilograms into grams by multiplying by 1000.

20.5 kg = 20,500 g

20500 ÷ 245 = 83.67…

Normally we would round 83.67… to 84, but in this case it is not practical as the last bag did not fill completely. Therefore the maximum number of bags that could be filled is 83.

Question

A hoist can lift 1,600 kg, given to 2 significant figures. It is being used to load boxes that weigh 48 kg to the nearest kg.

How many boxes can the hoist safely lift?

Solution

For a quick recap on significant figures see the section Significant figures.

For safety we need to think about the lower limit for the hoist and the higher limit for each box.

The lower limit for the hoist is 1,550 kg.

The higher limit for each box is 48.5 kg.

1550 ÷ 48.5 = 31.9….

Even though this is very close to 32, the hoist can only lift 31 boxes safely.

Algebraic Fractions

http://www.bbc.co.uk/education/guides/zx3cwmn/test

Trig Functions

http://www.bbc.co.uk/bitesize/higher/maths/revision_videos/trig_functions/video/

Vectors

Calculating the Modulus of a Vector

In this section, you will learn how to calculate the modulus of a vector. The modulus is a mathematical term for the length or the magnitude.

 

The magnitude of vector x is written as |x|.

The magnitude of vector   is written as |AB|.

 

 

Zero vector and unit vectors

A vector with magnitude 0 is called the zero vector, written 0. A vector with magnitude 1 is called a unit vector.

Vectors are equal if they have the same magnitude and the same direction.

a = b 

Inverse Vectors

The inverse of a vector is a vector of equal magnitude but in the opposite direction. The inverse of  is - or  and the inverse of a is -a.

Scalars

Scalars have magnitude but not direction. Vectors can be multiplied by a scalar to produce another vector.

Multiplying vector x by 3 will give a new vector 3 times the length and parallel to x.

Vector addition and subtraction

 

When 2 vectors are added or subtracted the vector produced is called the resultant.

The resultant is identified by a double arrowhead.

Triangle Law:

To add two vectors you apply the first vector and then the second.

 + = 

or

a + b = c

Subtracting a vector is the same as adding its inverse.

a – b is the same as a + (-b)

Parallelogram Law:

Moving from A to C through B is the same as moving through D.

 + = +  = 

or

a + b = b + a = c