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Further Maths

Chapter 3

Summarising Numerical Data

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The Mean : A measure of centre

• The mean (often referred to as the average).• Mean = sum of values

total number of values

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The mean - Demo

• Calculate the mean of the following numbers4 6 7 8 9 11

= 45 = 6

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The mean – Your Turn

• Calculate the mean of the following numbers14 20 25 30 32 133

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The mean – from Stem Plot

• 1 2 3 3 • 2 3 4 5 7 8 • 3 0 1 3 8 • 4 • 5• 6 0

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Median or Mean – Which one?

mean = median = mode =

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Median or Mean – Which one?

mean = median = mode =

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Median or Mean – Which one?

mean = median = mode =

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Median or Mean – Which one?

• When the data is symmetrical and there are no outliers, either the mean or the median can be used to measure the centre

• When the data is clearly skewed and/or there are outliers, the median should be used to measure the centre.

• Exercise 3A Pages 59-60 Questions 1-8

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• To measure the spread of data about the mean, the standard deviation is used.

• The formula for the standard deviation is

• Calculate the standard deviation for the following numbers 46 7 8 9 11

• Calculator display

The Standard Deviation : A measure of Spread about the Mean

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Estimating the Standard Deviation

• The standard deviation is approximately equal to the range divided by 4.

• Exercise 3B Pages 63-64 Questions 1 - 7

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The normal distribution

• Ball drop

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The 68%, 95%, 99.7% rule for normal distributions

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The 68%, 95%, 99.7% rule for normal distributions

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The 68%, 95%, 99.7% rule for normal distributions

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The 68%, 95%, 99.7% rule for normal distributions

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The 68%, 95%, 99.7% rule for normal distributions

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Example

• The heights of 200 boys was found to be normally distributed with a mean of 180 cm and a standard deviation of 10 cm.

• 68% of the heights will lie between• 95% of the heights will lie between• 99.7% of the heights will lie between

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Example

• What percentage of the heights will lie between 180 cm and 190 cm?

• What percentage of the heights will lie below 200 cm?

• What percentage of the boys will have a height between 160 cm and 190 cm?

• How many boys should have a height above 190 cm?

• Exercise 3C Pages 69-70 Questions 1-5

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Z scores ( standardised scores)

• Two students compare their scores for a test for two different subjects. One student scored 73 out of 100 for an English test. The other student scored 65 out of 100 for a Mathematics test. Which student performed better?

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Z scores ( standardised scores)The plot thickens

• Unbeknownst to the students the results for the English test were normally distributed with a mean of 80 and a standard deviation of 5. The results for the Mathematics test were normally distributed with a mean of 60 and a standard deviation of 8. Which student now performed better?

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Z scores

• Z scores are used to compare values from different distributions. A raw score, x, from a data set will have a z-score of (x – x )

swhere x is the mean of the data set and s is the standard deviation. The closer the z score is to zero, the closer the raw score is to the mean.

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Z scores

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Z scores

English z = (73 – 80) = - 7 = - 1.2 5 5

Maths z = (65 – 60) = 5 = 0.625 8 8

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Questions

• Exercise 3D Page 72 Questions 1-2

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Generating a random sample

• Draw numbers out of a hat• A spinner• Select every 2nd or 3rd person• Use technology

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Using your calculator to generate a simple random sample

• TI NSpire• Rand(25) will generate numbers from 1 to 25

• Calculator demonstration

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Displaying and calculating the summary statistics for grouped data using a calculator

Handspan (mm) Frequency

200- 1

210- 6

220- 8

230- 2

240- 1

250- 1

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