Kinds of data

10 red15 blue5 green

160cm172cm181cm

4 bedroomed3 bedroomed2 bedroomed

size 12size 14size 16size 18

fredlissymaxjack

callumzoeluke

stephen

baby 5lb3oz6lb10oz7lb12oz11lb1oz

Qualitative Continuous

Quantitative Discrete

Page 10Exercise 2AQ1 - 5 and 7

Averages

There are three types of average and they all begin with M

.....most popular value or class

.....middle value if all values are placed in order

.....the sum of all the values shared by how many values there are

MeanHow would you say the mean average differed

from the median average?In which circumstances may you use the mean

rather than the median and vice versa?

Rather than describing how to find the mean in words we need to learn some notation.

The sum of all the x values should be written as:

The sum of the values in the fx column should be written as:

Page 15Exercise 2B

Q1, 2, 4, 6 and 7

In what kind of questions will we need to add the values in the fx column rather than just add together all the x values?

Try finding the mean of the year 7 girl heights and compare it to the year 7 boy

heights.Would you use the mean or the median to

summarise this data and why?

If you used the raw data for the above calculation why?

If you used the grouped data from the frequency polygon or histogram work why

am I going to tell you you've made a mistake?

We use the "x Bar" notation to represent a sample mean.

If I was using all the data possible it would be called a population

mean and we use the "mu" notation

Mean Formulae

for a list of data

for a frequency table of data

for grouped data it is necessary to find the midpoint of each class first and use this as a value for x and then use the same equation above.

Page 18Exercise 2CQ1, 4 and 5

Stem and Leaf

Put the data below into a suitable stem and leaf diagram.

127, 135, 147, 147, 149, 139, 145, 155, 149, 155, 151, 159, 139, 141, 155, 160, 138, 144, 155, 148156, 143, 147, 157, 152, 150, 161, 133, 146, 155

The data represents heights of a first year class in a boys

school.

How else can you summarise or represent this data?

Below is a list of the heights of 30 year 7 girls. Add these to the other side of your

stem and leaf diagram and make some comparitive statements based on suitable

summary data you find.

127, 145, 147, 147, 149, 149, 145, 165, 139, 157, 152, 169, 129, 121, 158, 160, 148, 141, 155, 148156, 143, 157, 156, 152, 150, 161, 133, 146, 155

Page 55Exercise 4A

Q2 and 5

If you have grouped your data with equal class widths, you have little to worry

about. However if the class widths are uneven you will need to plot them against

frequency density rather than just frequency.

Frequency Density = Class Frequency

Class Width

Sometimes Relative Frequency Density is plotted on the y axis. This can be

calculated as:

Rel Freq Dens = Class Frequency Total

Frequency

It is best that Histograms are plotted against frequency density or relative

frequency density.

They should also only be drawn with continuous data.

Discrete or qualitative data can be plotted in Bar Charts but their bars should not really touch as they aren't connected

Histograms

Histograms are similar to bar charts apart from the consideration of areas.

In a bar chart, all of the bars are the same width and the only thing that matters is the

height of the bar. In a histogram, the area is the important

thing.

Page 64Exercise 4EQ1, 4 and 5

We may also need to find the average from these

grouped continuous data sets

Page 22Exercise 2D

Q1 - 5

Summarising Data

What types of data summary have you come up with so far?

And how do they differ?Give examples of when one type would be better

than another.

Quartiles

You will recall finding the median from a list of data involves adding one to the number of values before halving it to find out which value (placed in order) you should use.

For example in a list of 7 numbers the median value is the(7 + 1) / 2 th valuethe 4th value

3, 5, 5, 6, 8, 10, 10

If you consider the quartiles you can see that it is the 2nd and 6th values. These can be

found by dividing the number of values by 4 and as long as this gives a whole number find the average of it's value and the

value above it.If this yields a decimal value rather than a whole number then

always round UP to the value above it.In a list of 14 numbers the lower quartile would be taken as the

14 / 4 th value or 3.5th valueso you'd take the 4th value

1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14

Page 34Exercise 3A

Q1 and 2

Interquartile Ranges and Boxplots

We use the Quartiles and minimum and maximum values to draw boxplots.

These are great for comparing the spread of data between two or more data sets

Q1 Q3Q2

Q0 Q4

Where Q0 = min valueQ1 = lower quartileQ2 = medianQ3 = upper quartileQ4 = max value

Draw box plots to compare the year 7 height data you put into stem and

leaf diagrams earlier

Read page 57Page 58

Exercise 4BQ1 - 2

Page 59Exercise 4C

Q1 - 2

Page 61Exercise 4D

Q1 - 2

Cumulative Frequency

We have seen how to find the quartiles from a list of data or stem and leaf diagrams.

We have also seen that data is often stored in frequency distributions. If these are grouped it

becomes difficult to find these quartiles. Why?

We used to overcome this by drawing cumulative frequency curves.

60 students got below 50 marks

30 students got below 40 marks

by joining up these two points we pretend the 30 students with between 40 to 50 marks are spread evenly throughout the band.

Interpolation

We could actually find this value much quicker by using some simple mathematics known as

interpolating.

Think how you could find the mark of the 40th student from this year 10 class using just the

data rather than reading from the graph.

Now try estimating the mark of the 55th student.

Remember estimating doesn't mean guessing; it involves exact calculations but it is unlikely to be the true mark as the students are unlikely to be spaced

evenly througout the class. we cannot know the exact mark without the raw data - we are not given this with

grouped data - hence we estimate

Quantile = b + (Qn - f) x wfm

Where fm is the frequency of the class the quantile falls in

f is the cumulative frequency up to the class the quantile falls in

w is the class width of the class the quantile falls inQ is the quantile you are finding expressed as a

fractionn is the number of data

b is the LOWER bound of the class the quantile fits in

We can divide the data into as many equal parts as we like.

Quartiles divide in fourDeciles in ten and Percentiles into 100

The formula below is known as interpolating and estimates a quantile by assuming the data

collected in each class is spread evenly.It should be very similar to the formula yo

created earlier to find quartiles without drawing a cumulative frequency curve.

Quantiles

Page 34Exercise 3A

Q3 and 5

Page 37Exercise 3BQ1, 3 and 5