Transcript Title goes here - St Pats Mr Anthony Nunan

Further Maths
Chapter 3
Summarising Numerical Data
1
The Mean : A measure of centre
• The mean (often referred to as the average).
• Mean =
sum of values
total number of values
2
The mean - Demo
• Calculate the mean of the following numbers
4
6
7
8
9
11
= 45
=6
7.5
3
The mean – Your Turn
• Calculate the mean of the following numbers
14
20
25
30
32
133
4
The mean – from Stem Plot
•
•
•
•
•
•
1
2
3
4
5
6
233
34578
0138
0
5
Median or Mean – Which one?
mode =
median =
mean =
6
Median or Mean – Which one?
mode =
median =
mean =
7
Median or Mean – Which one?
mode =
median =
mean =
8
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
9
The Standard Deviation :
A measure of Spread about the
Mean
• To measure the spread of data about the mean, the
standard deviation is used.
• The formula for the standard deviation is
2
• Calculate the standard deviation for the following
numbers 4 6 7 8 9 11
• Calculator display
10
Estimating the Standard
Deviation
• The standard deviation is approximately equal
to the range divided by 4.
• Exercise 3B Pages 63-64 Questions 1 - 7
11
The normal distribution
• Ball drop
12
The 68%, 95%, 99.7% rule for
normal distributions
13
The 68%, 95%, 99.7% rule for
normal distributions
14
The 68%, 95%, 99.7% rule for
normal distributions
15
The 68%, 95%, 99.7% rule for
normal distributions
16
The 68%, 95%, 99.7% rule for
normal distributions
17
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
18
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
19
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?
20
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?
21
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 )
s
where 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.
22
Z scores
23
Z scores
English z = (73 – 80) = - 7 = - 1.2
5
5
Maths z = (65 – 60) = 5 = 0.625
8
8
24
Questions
• Exercise 3D Page 72 Questions 1-2
25
Generating a random sample
•
•
•
•
Draw numbers out of a hat
A spinner
Select every 2nd or 3rd person
Use technology
26
Using your calculator to generate a
simple random sample
• TI NSpire
• Rand(25) will generate numbers from 1 to 25
• Calculator demonstration
27
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
28
29
30