Transcript PROBLEM This is where you decide what you would like more information on. CONCLUSION What did you learn about your investigation • What do the graphs.

PROBLEM
This is where you decide
what you would like more
information on.
CONCLUSION
What did you learn about your investigation
• What do the graphs say
• What differences are there in the
statistics
• Can you infer that the difference in the
sample is also in the population
• Are there new problems to investigate
PLAN
You need to know what
you will measure and how
you will do it.
• What data do you need
• How will you collect it
• What will you record
• How will you record it
ANALYSIS
This is where you look at the data to
see what it tells you about your
problem.
• Graph data and collect statistics
DATA
This is where your data is
collected, managed and
organised.
What is it that you would like information on, considering information is
often more interesting when one group is different to another.
Remember that when comparing groups we should look to use numeric
data or qualitative groups.
Once you have decided what you would like information on you need to
write a question fully describing what you are interested in investigating.
There are two ways in which you can write a question. Two example
questions investigating bag weights could be:
e.g. ‘I wonder if there is a difference between the bag weights of Year 11
boys and Year 11 girls at CHS’
e.g. ‘’Do Year 11 boys at CHS tend to have heavier bag weights
than Year 11 girls at CHS’
Now that you know what you are going to investigate, you will need to
find variables that will illustrate the differences.
Sometimes there are restrictions on what data can be collected or the
ease with which data can be collected.
You may need to consider how you will collect and record the data.
Data is often best recorded in a table.
There are web sites from which you can extract data and a good
starting point is asking yourself what differences you will expect.
Remember: This Achievement Standard is an investigation into
multivariate data which implies that you will initially look at a number of
variables before deciding which you will look for differences between
It is important that the largest sample that is possible is used. Recommended
minimum is 30 as drawing conclusions from small samples is suspect.
Census At School is an excellent resource that contains a huge amount
of interesting statistical information. It is likely this data will be used.
When you collect data from website, you may need to ‘clean’ it. This
refers to the process of removing invalid data points from your sample.
e.g. When using data regarding bag weights, if an individual has no bag,
do not record it as a 0.
e.g. Check units are consistent, all 0 entries and always be suspicious of
any data that seems out of place.
MEASURES OF CENTRAL TENDENCY
1.
Mean
- easy to calculate but is affected by extreme values
- to calculate use:
Sum of all values
Total number of values
e.g. Calculate the mean of 6, 11, 3, 14, 8
Mean =
6 + 11 + 3 + 14 + 8
=
5
42
Push equals on
calculator BEFORE
dividing
= 8.4
5
e.g. Calculate the mean of 6, 11, 3, 14, 8, 100
Mean =
6 + 11 + 3 + 14 + 8 + 100
6
=
142
6
= 23.7 (1 d.p.)
2.
Median
- middle number when all are PLACED IN ORDER (two ways)
- harder to calculate but is not affected by extreme values
a) for an odd number of values, median is the middle value
e.g. Find the median of 39, 44, 38, 37, 42, 40, 42, 39, 32
32, 37, 38, 39, 39, 40, 42, 42, 44
To find placement of
median use:
9 + 1 = 10 = 5
n+1
2
2
2
Median = 39
n = amount of data
Cross of data, one at a
time from each end until
you reach the middle
value.
b) for an even number of values, median is average of the two middle values
OR
e.g. Find the median of 69, 71, 68, 85, 73, 73, 64, 75
64, 68, 69, 71, 73, 73, 75, 85
n + 1 = 8 + 1 = 4.5
2
2
Median = 71 + 73 = 144 = 72
2
2
OR
3.
Mode
- only useful to find most popular item
- is the most common value (can be none, one or more)
e.g. Find the mode of 188, 93, 4, 93, 15, 0, 100 15
Mode = 15 and 93
MEASURES OF SPREAD
Range
- can show how spread out the data is
- is the difference between the largest and smallest values
e.g. Find the range of 4, 2, 6, 9, 8
lowest value
highest value
Range = 9 – 2
= 7 (2 – 9)
Note: Its a good idea to write in brackets the values that make up the range.
Standard Deviation
– is the measure of the average spread of the numbers from the mean.
– for Year 11, your only concern is that the bigger the value, the more spread
the data is.
Quartiles
– are measures of spread which with the median splits the data into quarters
– method used is similar as to when finding median
When the data is in order:
– the lower quartile (LQ) has 25% or ¼ of the data below it.
– the upper quartile (UQ) has 75% or ¾ of the data below it.
– the Interquartile Range (IQR) = UQ – LQ and describes the middle 50%
¼
e.g.
Find the LQ, UQ and the interquartile range of the following data
6, 6, 6, 7, 8, 9, 10, 10, 11, 14, 16, 16, 17, 19, 20, 20, 24, 24, 25, 29
Note: always find the median first
10 + 1 = 11 = 5.5
2
2
20 + 1 = 21 = 10.5
or cross off data
2
2
LQ =8 + 9 = 17 = 8.5 OR UQ = 20 + 20 = 40 = 20 OR
2
2
2
2
IQR = 20 – 8.5 = 11.5
IQR = UQ - LQ
e.g.
Find the LQ, UQ and the interquartile range of the following data
5, 6, 8, 10, 11, 11, 12, 15, 18, 22, 23, 28, 30
Remember, always find the median first
13 + 1 = 14 = 7
or cross off data
2
2
As the median is an actual piece of data, it is ignored when finding the LQ and UQ
6 + 1 = 7 = 3.5
2
2
LQ = 8 + 10 = 18 = 9
2
2
IQR =
UQ = 22 + 23 = 45 = 22.5
2
2
22.5 – 9 = 13.5
Dot Plots
– are like a bar graph
– each dot represents one item
e.g.
Plot these 15 golf scores on a dot plot
70, 72, 68, 74, 74, 78, 77, 70, 72, 72, 76, 72, 76, 75, 78
Range plot between lowest
and highest values
68
70
72
Golf Scores
74
76
78
Stem and Leaf Plots
– records and organises data
– most significant figures form the stem and the final digits the leaves
– can be in back to back form in order to compare two sets of data
e.g.
Place the following heights (in m) onto a back to back stem and leaf plot
BOYS = 1. 59, 1.69, 1.47, 1.43, 1.82, 1.70, 1.73, 1.35, 1.76, 1.68,
1.62, 1.84, 1.45, 1.50, 1.54, 1.73, 1.84, 1.71, 1.66
GIRLS = 1. 44, 1.46, 1.63, 1.29, 1.48, 1.57, 1.51, 1.42, 1.34, 1.45,
1.57, 1.59, 1.42
Look at the highest and lowest data values to decide the range of the stem
Unordered Graph of Heights
Ordered Graph of Heights
Boys
Girls
Boys
Girls
4 ,4 ,2 1.8
4, 4, 2 1.8
1 ,3 ,6 ,3 ,0 1.7
6, 3, 3, 1, 0 1.7
6 ,2 ,8 ,9 1.6 3
9, 8, 6, 2 1.6 3
4 ,0 ,9 1.5 7, 1, 7, 9
9, 4, 0 1.5 1, 7, 7, 9
5 ,3 ,7 1.4 4, 6, 8, 2, 5, 2
7, 5, 3 1.4 2, 2, 4, 5, 6, 8
5 1.3 4
5 1.3 4
1.2 9
1.2 9
Place the final digits of the data on the graph on the correct side
Calculating Statistics from Stem and Leaf Plots
For each statistic, make
Graph of Heights
sure to write down the
Boys
Girls
whole number, not just
4, 4, 2 1.8
the ‘leaf’!
6, 3, 3, 1, 0 1.7
9, 8, 6, 2 1.6 3
9, 4, 0 1.5 1, 7, 7, 9
7, 5, 3 1.4 2, 2, 4, 5, 6, 8
5 1.3 4
1.2 9
When finding median, LQ
and UQ, make sure you
count/cross in the right
direction!
e.g. From the ordered plot state the minimum, maximum, LQ, median, UQ, IQR
and range statistics for each side
Minimum:
Maximum:
LQ:
Median:
UQ:
IQR:
Range:
BOYS
1.35 m
1.84 m
1.50 m
1.68 m
1.73 m
1.73 – 1.50 = 0.23 m
1.84 – 1.35 = 0.49 m
GIRLS
Median = 13 + 1 = 7
1.29 m
2
1.63 m
1.42 m
LQ/UQ = 6 + 1 = 3.5
1.46 m
2
1.57 m
1.57 – 1.42 = 0.15 m
1.63 – 1.29 = 0.34 m
Remember: If you find it hard to calculate stats off graph, write out data in a line first!
Box and Whisker Plots
– shows the minimum, maximum, LQ, median and UQ
– ideal for comparing two sets of data
e.g.
Note: Use the minimum
and maximum values
to determine length of
scale
Using the height data from the Stem and Leaf diagrams, draw two box and
whisker plots (Boys and Girls)
Box and Whisker Plot of Boys and Girls Heights
Males
Minimum
LQ
Median
UQ
Maximum
Females
1.20
1.30
1.40
1.50
1.60
1.70
Height (m)
Question: What is the comparison
between the boy and girl heights?
ANSWER?
EVIDENCE?
1.80
1.90
)
d
Graph the grouped frequency table data about heights onto a histogram
n
e.g.
c
e
y
n
t
s
Histograms
– display grouped data
– frequency is along vertical axis, group intervals are along horizontal axis
– there are NO gaps between bars
e
t
u
s
u
Note: The groups from the table form the intervals along the horizontal axis and
the highest frequency determines the height of the vertical axis.
Student Heights
q
f
e
o
12
10
.
r
8
o
F
6
(
n
4
2
0
140
150
160
Height (cm)
170
180
190
– Side by side histograms can also be used to compare data
Female Heights
Male Heights
8
8
6
6
4
4
2
2
140
150
160
170
180
190
200
140
150
160
170
180
190
200
Question: What is the comparison between the female and male heights?
ANSWER?
EVIDENCE?
g
k
(
Scatter Plots
– looks for a relationship between two measured variables
– points are plotted like co-ordinates
Use the data to
determine scale to
use on both axes
Scatte r Di a gra m fo r boys hei g hts and wei g hts
g
60
i
Weight
(kg)
48
52
50
49
53
47
58
45
50
51
49
46
44
49
Line of
best fit
55
e
Height
(cm)
144
152
161
148
155
140
158
139
147
150
152
138
137
145
W
e.g.
h
t
Outliers can
generally be ignored
Below are the heights and weights of Year 7 boys. Place on a scatter plot.
50
45
135
140
145
Hei g ht (c m)
150
155
If points form a line (or close to) we
can say there is a relationship
What is the relationship between the
between the two variables.
boys height and weight?
160
ANSWER?
EVIDENCE?
Time Series
– a collection of measurements recorded at specific intervals where the quantity changes with time.
)
Features of Time Series
a) Order is important with all measurements retained to examine trends
b) Long term trends where measurements definitely tend to increase or decrease
c) Seasonal trends resulting in up and down patterns What are the short and long term
trends? ANSWER? EVIDENCE?
e.g. Draw a time series graph for the following data:
(
9900
9500
e
s
9700
l
9300
a
9100
9
8
9
7
9
6
9
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8300
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8500
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Join up each of the points
8700
9
8900
S
Sept.
Dec.
Mar. 96
June
Sept.
Dec.
Mar. 97
June
Sept.
Dec.
Mar. 98
June
Sept.
Dec.
Mar. 99
Quarterly
sales
9040
8650
8370
9250
9033
8578
8495
9407
9209
8740
8618
9504
9246
8929
8670
S
Season
$
Quarterly Sales for Elliots's Fish and Chips Shop
Quarter Years
When you have finished your analysis, it is important to see if you can make an
inference. An inference is when you make a generalised statement about the
whole population (generally if there is a difference), by using the findings from
your sample.
It is easiest if you write your statement in the following manner:
“From my sample data I can make an inference about the population that…
Year 11 Boys tend to be taller than Year 11 Girls (example). This is because…
You will need to justify this inference by using your findings from both your
statistics and/or graphs. Ways to justify your inference will be shown on an
extra handout.
Once you have made your inference with justifications, you should end off
your investigation by making a conclusion. Your conclusion MUST answer
your original question and can be written in the format:
“Therefore I can conclude that… Year 11 boys at CHS are typically taller
than Year 11 girls at CHS (example).
A sample should:
1) Be large enough to be representative of whole population
2) Have people/items in it that are representative of the population
It is best to choose samples that are large and random but size may be
affected by time, money, personnel, equipment etc.
Simple random sampling:
1- obtain a population list
2- number each member
3- use random table or random number on calculator
Systematic sampling:
1- obtain a population list
2- randomly select a starting point on the list
3- select every nth member until desired sample size is reached
Note: every nth member is found by: Population/group size
Size of sample needed
1. In terms of Data Collection
Typical Limitations
- Sample too small
- Not random or
representative
- Taken over too short
a time period
- Outliers distort data
2. In terms of Your Process
Typical Limitations
- Not enough statistics
calculated
- Not enough graphs used,
data could be compared better
- Scales on graphs too large
- Way graphs are drawn
Improvements
- Obtain a bigger sample
- Get a representative
sample
- Take data over a longer
time period
- Ignore extreme outliers
Improvements
- Calculate more statistics
- Use other graphs (i.e.
comparative histograms)
- Change scales on graph
(smaller)
- Alter the way the graphs may
be drawn