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### Note 10: Box and Whisker Plots

Statistics

Year 9

Note 1: Statistical Displays

Note 1: Statistical Displays

Note 1: Statistical Displays

Note 1: Statistical Displays

Dot Plots - Symmetry

A symmetric distribution can be divided at the centre so that each half is a mirror image of the other.

Dot Plots - Skewness

Dot Plots - Outliers

A data point that diverges greatly from the overall pattern of data is called an outlier.

Dot Plots

e.g. This graph shows the number of passengers on a school mini bus for all the journeys in one week.

IWB Ex31.02

Pg 863

19

How many journeys were made altogether?

What was the most common number of passengers?

6

Note 3: Pie Graphs

Pie Graphs are used to show comparisons

‘Slices of the Pie’ are called sectors

Skills required: working with percentages & angles

e.g. 20 students in 9Ath come to school by the following means:

10 walk

5 Bus

3 Bike

2 Car

Represent this information on a pie graph.

Note 3: Pie Graphs

e.g. 20 students in 9Ath come to school by the following means:

10 walk

5 Bus

3 Bike

2 Car

= 10 × 18°

= 180°

= 90°

= 5 × 18°

= 54°

= 3 × 18°

= 36°

= 2 × 18°

All 20 Students represent all 360°of a pie graph

How many degrees does each student represent?

= 18°

Note 3: Pie Graphs

IWB Ex31.03

Pg 870

We can also use percentages and fractions to calculate the angles

e.g. 500 students at JMC were surveyed regarding their TV provider at home. 180 had Skyview, 300 had Freeview and 20 had neither. Represent this in a pie chart.

× 360°

= 129.6°

× 360°

= 216°

× 360°

= 14.4°

Note 4: Stem & Leaf Graphs

Daily absences from JMC for a six week period in Term 3 are as follows:

Note 4: Stem & Leaf Graphs

Daily absences from JMC for a six week period in Term 3 are as follows:

These figures can be summarized in a stem and leaf graph

Note 4: Stem & Leaf Graphs

Note 5: Scatter Plot

IWB Ex31.05

Pg 879

Eg: this has a positive relationship – the taller the person the longer they can jump

Scatter Plots show the relationship between two sets of data.

Note 6: Time Series Graph

This ‘line graph’ shows what happens to data as time changes

Time is always on the x-axis

Data values are read from the y axis

What are some of the features of this graph?

# of advertisements

Time

Note 6: Time Series Graph

Each week, roughly the same amount of advertisements are sold

The most popular days to advertise are:

Wednesday & Saturday

The least popular days to advertise are:

Monday & Tuesday

What are some of the features of this graph?

IWB Ex31.06

Pg 884

IWB Ex31.05

Pg 879

Calculating Statistics - averages

- Mean (average) – The mean can be affected by extreme values
- Median – middle number, when all data is placed in order. Not affected by extreme values
- Mode – the most common value/s

Note 7: Median

- Median – middle number, when all the number are placed in order. Not affected by extreme values

Note 7: Median

- Median – middle number, when all the number are placed in order. Not affected by extreme values

Note 7: Mode

- Mode – is the most common value, one that occurs most frequently

e.g. Find the mode of the following

Note 7: Calculating Averages

- In statistics, there are 3 types of averages:
- mean
- median
- mode

Mode

Median

Mean - x

The middle value when all values are placed in order

The most common value(s)

Affected by extreme values

Not Affected by extreme values

IWB Ex31.07 Pg 892

Ex31.08 Pg 896

Ex31.09 Pg 901

Starter

- Calculate the mean for each of the following:
a) 4, 8, 12, 4, 1, 1

b) 40, 50

c) 21, 0, 19, 20

- Ten numbers add up to 89, what is their mean
- Calculate the mean to 2dp
a) 84, 31, 101, 6, 47, 89, 49, 55, 111, 39, 98

b) 1083, 417, 37.8, 946

- A rowing ‘eight’ has a mean weight of 86.375kg. Calculate their combined weight
- A rugby pack of 8 schoolboy players with a mean weight of 62kg is pushing against a pack of 6 adult players with a mean weight of 81kg. Which pack is heavier? Explain why?

Note 8: Frequency Tables

A frequency table shows how much there are of each item. It saves us having to list each one individually.

8

2

4

56, # of houses

Note 8: Frequency Tables

How would you display this information in a graph?

Note 8: Frequency Tables

Tables are efficient in organising large amounts of data. If data is counted, you can enter directly into the table using tally marks

e.g 33 students in 10JI were asked how many times they bought lunch at the canteen. Below is the tally of individual results.

0 4 0 3 5 0 5 5 0 2 1

0 5 2 3 0 0 5 5 1 2 5

5 3 0 0 1 5 0 5 1 3 0

The data can be summarised in a frequency table

Note 8: Frequency Tables

IWB Ex31.11

Pg 910

Calculate the mean =

=

=

= 2.3

Why is this mean misleading?

Most students either do not buy their lunch at the canteen or buy it there every day.

Total 33

Starter

- Write down the median of each of these sets of numbers
a) {12, 19, 22, 28, 31}

b) {0, 6, 9, 11, 19, 20}

- Write down the mode foe each of these sets of numbers
a) {6, 8, 9, 9, 10, 6, 7, 9, 8}

b) {4, 6, 8, 6, 4, 8}

c) {3, 1, 0, 1, 5, 0, 6}

- A roadside stall has some avocados for sale at $2 a bag. These are the coins in the ‘honesty’ box on Tuesday.
5 x 20c coins 2 x 50c coins 2 x $1 coins 1 x $2 coins

a) what is the median of the coins

b) on Wednesday there were 24 coins in the box. The mean value of the coins was 25cents. Which gives better information about the number of bags sold – the mean or the median.

22

10

9

No mode

Two modes are 0 and 1

35 cents

The mean gives information about the total sold: 24 x 25cents = $6. 3 bags were sold

Note 9: Histograms

When a frequency diagram has grouped data we use a histogram to display it

- measured data (e.g. Height, weight)

Note 9: Histograms

When a frequency diagram has grouped data we use a histogram to display it

Note 9: Histograms

IWB Ex31.12

Pg 916

When a frequency diagram has grouped data we use a histogram to display it

Calculating Statistics

- Range – a measure of how spread out the data is. The difference between the highest and lowest values.
- Lower Quartile (LQ) – halfway between the lowest value and the median
- Upper Quartile (UQ) – halfway between the highest value and the median
- Interquartile Range (IQR) – the difference between the LQ and the UQ. This is a measure of the spread of the middle 50% of the data.

Example:

e.g. 40, 41, 42, 43, 44, 45, 49, 52, 52, 53

UQ

median

LQ

Range = Maximum – Minimum

= 53 – 40

= 13

IQR (Interquartile Range) = UQ – LQ

= 52 – 42

= 10

The five key summary statistics are used to draw the plot.

Minimum LQ Median UQ Maximum

Note 10: Box & Whisker Plot

Comparing data

Male

Female

x

median

minimum

maximum

Upper

quartile

extreme value

Lower

quartile

IQR

e.g.

1 1 2 2 3 3 4 4 4 5 6 7 18

LQ

UQ

The following data represents the number of flying geese sighted on each day of a 13-day tour of England

5 1 2 6 3 3 18 4 4 1 7 2 4

Find:

a.) the min and max number of geese sighted

b.) the median

c.) the mean

d.) the upper and lower quartiles

e.) the IQR

f.) extreme values

Min – 1

Max - 18

Order the data - 4

Add all the numbers and divide by 13 – 4.62 (2 dp)

LQ – 2 + 2 = 2

UQ – 5 + 6 = 5.5

2

2

5.5 – 2 = 3.5

18

Note 11: Quartiles

e.g. Calculate the median, and lower and upper quartiles for this set of numbers

35 95 29 95 49 82 78 48 14 92 1 82 43 89

Arrange the numbers in order

1 14 29 35 43 48 49 78 82 82 89 92 95 95

LQ

UQ

median

Median – halfway between 49 and 78, i.e. = 63.5

LQ – bottom half has a median of 35

UQ – top half has a median of 89

Summary: Data Display

Line Graphs – identify patterns & trends over time

Interpolation -

Reading in between tabulated values

Extrapolation -

Estimating values outside of the range

Looking at patterns and trends

0 1 2 3 4 5 6 7 8 9 10 11

Summary: Data Display

Pie Graph – show proportion

Multiply each percentage of the pie by 360°

60% - 0.6 × 360° = 216°

Scatter Graph – show relationship between 2 sets of data

Plot a number of coordinates for the 2 variables

Draw a line of best fit - trend

Reveal possible outliers (extreme values)

Summary: Data Display

Histogram– display grouped continuous data

– area represents the frequency

frequency

Bar Graphs– display discrete data

Distance (cm)

– counted data

– draw bars (lines) with the same width

– height is important factor

Summary: Data Display

Stem & Leaf – Similar to a bar graph but it has the individual numerical data values as part of the display

– the data is ordered, this makes it easy to locate median, UQ, LQ

3 3 4 8

5

10

9 8 8 3

11

2 3 6 7 8

Back to Back Stem & Leaf – useful to compare spread & shape of two data sets

4 2 0

12

1 9 9

3 3

13

0 2

2

14

5

Key: 10 3 means 10.3

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