GCSE MATHS

GCSE Session 28 - Cumulative Frequency, Vectors and Standard Form

Probability from last weekWhen the question asks for ‘and’ we need to multiply both probabilities. The two events MUST BE INDEPENDENT. If they are not Independent then we need to create a tree diagram, we will still multiply but the number used are dependent on each other.

To solve the problem I had last week

An example of INDEPENDENT events where we just multiply the separate probabilities is:

Chance of getting a 6 and a head when rolling a dice and tossing a coin.

1/6 x ½ = 1/12

Consider outcomes when tossing two coins (independent) Head Head ½ x ½ = 1/4 Head Tail ½ x ½ = 1/4 Tail Head ½ x ½ = 1/4 Tail Tail ½ x ½ = 1/4

P(both same) = HH or TT = P(HH) + P(TT)

P(both different) = HT or TH = P(HT) + P(TH)

You will need graph paper for plotting graphs.

Cumulative Frequency – (Higher)

The cumulative frequency is obtained by adding up the frequencies as you go along, to give a 'running total'.

The table shows the lengths (in cm) of 32 cucumbers. Before drawing the cumulative frequency diagram, we

need to work out the cumulative frequencies. This is done by adding the frequencies in turn.

Length Frequency Cumulative Frequency

21-24 3 3

25-28 7 10 (= 3 + 7)

29-32 12 22 (= 3 + 7 + 12)

33-36 6 28 (= 3 + 7 + 12 + 6)

37-40 4 32 (= 3 + 7 + 12 + 6 + 4)

The points are plotted at the upper class boundary. In this example, the upper class boundaries are 24.5, 28.5, 32.5, 36.5 and 40.5.

Cumulative frequency is plotted on the vertical axis.

Both sets of numbers are always increasing

There are 32 cucumbers. When we have 32 pieces of data, the median is halfway between the 16th and 17th.

We could call this this 16 ½th value.

For ‘n’ numbers, the rule for finding the median is: ½ (n +1)

So 32 + 1 = 33. ½ of 33 = 16 ½

The range of a set of data is between the largest and the smallest

The median cuts the data in half, half is larger and half smaller.

It can be useful to split the data into quarters, we do this in a similar way to dining the median, but use ¼ and ¾ instead of ½.

Once we have done this the data is in 4 quarters. Half the data is contained within the second and third quarters.

The range between the lower quartile and the upper quartile is known as the interquartile range.

This is a more useful measure of spread because it if no influenced by extreme values.

Calculating the upper and lower quartiles

For ‘n’ numbers, the rule for finding the Lower Quartile is: ¼ (n +1)

For ‘n’ numbers, the rule for finding the Upper Quartile is: ¾ (n +1)

¼ (33) = 8.25 ¾ (33) = 24.75

Note for large frequencies, sometimes the +1 is not always used, just ‘n’.

Interquartile range

A better way to measure spread than the traditional ‘range’ Is the interquartile range. i.e. the difference between the upper quartile and the lower quartile.

This measures the spread of the middle 50% of the data, and avoids outliers (extreme values)

Quartiles are a useful measure of spread because they are much less affected by outliers or a skewed data set than the equivalent measures of mean and standard deviation.(SD is an A level topic, but is another way of measuring spread)

For this reason, quartiles are often reported along with the median as the best choice of measure of spread and central tendency, respectively, when dealing with skewed and/or data with outliers.

A common way of expressing quartiles is as an interquartile range. The interquartile range describes the difference between the third quartile (Q3) and the first quartile (Q1), telling us about the range of the middle half of the scores in the distribution.

Plotting the curve When we plot the curve for cumulative

frequency, we plot against the upper limit of the group.

If there is a gap between groups (see example page 432) we take the class boundary to be half way between the top of one group and the bottom of the next.

Ex 39.1 Q1 then EX39.2 Q1

Comparing Distributions

We can generate the median and compare the average from one set of data to another.

By generating the IQR, we can say which data is more spread (larger IQR) and which is less spread (smaller IQR)

Ex39.3 Q3 Ext Q1

Box Plots Box plots, or box

and whisker diagrams allow us to see the min value, LQ, median, UQ and the max value. And from them we can read the range, interquartile range and median.

Example

The oldest person in Mathsminster is 90. The youngest person is 15. The median age of the residents is 44, the lower quartile is 25, and the upper quartile is 67.

Represent this information with a box-and-whisker plot. (you still need an axis or number line for this representation)

Compare the data above, what conclusions can we draw?

Ex 39.4 Q1 and 6 Extension – Review exercise 39

Standard Form

Numbers containing lots of zeros can express very large values and very small values.

Standard form is a way to abbreviate these numbers, but it follows very specific rules.

The first part is a number equal to or bigger than 1, but smaller than 10

1 ≤ n < 10 (including decimals)

This number is multiplied by a power of 10

n x 10a

Positives and negatives

6.2 x 105 = 620 000 You move the decimal place to the right

5 times.

3.7 x 10-4 = 0.00037 You move the decimal place to the left 4

times

Ex 6.1 Q3 and 4 then Ex 6.2 Q3 and 4

You must learn how to enter powers into your calculator, or if there is a function for entering x 10a

For non calculator questions you might write out the numbers with the zeros, do the calculation and then but the answer back in standard form. (see examples p61)

When calculating with numbers in standard form, they follow the same rules as all indices (p49 and 50)

We have covered the basics of standard form, use the review exercise 6 or BKSB to practice using and working with it.

Vectors

We have looked at these during transformations.

Read the what you need to know section of chapter 33 (p361)

Then attempt exam questions from review exercise 33 (p362)