Statistics

1
Statistics
S5 Int2
Quartiles from a Frequency Table
Quartiles from a Cumulative Frequency Table
Estimating Quartiles from C.F Graphs
Standard Deviation
Standard Deviation from a sample
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Scatter Graphs
Probability
Relative Frequency Probability
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Starter Questions
S5 Int2
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3
Statistics
Quartiles from Frequency Tables
S5 Int2
Learning Intention
Success Criteria

1. Know the term quartiles.

1. To explain how to calculate quartiles from
   frequency tables.

1. Calculate quartiles given a frequency table.

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4
Statistics
Quartiles from Frequency Tables
S5 Int2
Reminder !
Range The difference between highest and
Lowest values. It is a measure of spread.
Median The middle value of a set of
data. When they are two middle values the
median is half way between them.
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Mode The value that occurs the most in a set
of data. Can be more than one value.
Quartiles The median splits into lists of equal
length. The medians of these two lists are
called quartiles.
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Statistics
Quartiles from Frequency Tables
S5 Int2
To find the quartiles of an ordered list you
consider its length. You need to find three
numbers which break the list into four smaller
list of equal length.
Example 1 For a list of 24 numbers, 24 6 4
R0
6 number
6 number
6 number
6 number
Q1
Q2
Q3
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The quartiles fall in the gaps between Q1 the
6th and 7th numbers Q2 the 12th and 13th
numbers Q3 the 18th and 19th numbers.
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Statistics
Quartiles from Frequency Tables
S5 Int2
Example 2 For a list of 25 numbers, 25 4 6
R1
1 No.
6 number
6 number
6 number
6 number
Q1
Q3
Q2
The quartiles fall in the gaps between Q1 the
6th and 7th Q2 the 13th Q3 the 19th and 20th
numbers.
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Statistics
Quartiles from Frequency Tables
S5 Int2
Example 3 For a list of 26 numbers, 26 4 6
R2
6 number
6 number
6 number
6 number
Q2
1 No.
1 No.
Q1
Q3
The quartiles fall in the gaps between Q1 the
7th number Q2 the 13th and 14th number Q3 the
20th number.
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Statistics
Quartiles from Frequency Tables
S5 Int2
Example 4 For a list of 27 numbers, 27 4 6
R3
6 number
6 number
6 number
6 number
1 No.
1 No.
1 No.
Q2
Q1
Q3
The quartiles fall in the gaps between Q1 the
7th number Q2 the 14th number Q3 the 21th
number.
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Statistics
Quartiles from Frequency Tables
S5 Int2
Example 4 For a ordered list of 34. Describe
the quartiles.
34 4 8 R2
Q2
8 number
8 number
8 number
8 number
1 No.
1 No.
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Q1
Q3
The quartiles fall in the gaps between Q1 the
9th number Q2 the 17th and 18th number Q3 the
26th number.
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Statistics
Quartiles from Frequency Tables
S5 Int2
Now try Exercise 1 Start at 1b Ch11 (page 162)
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11
Starter Questions
S5 Int2
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Statistics
S5 Int2
Quartiles from Cumulative Frequency Table
Learning Intention
Success Criteria

1. Find the quartile values from Cumulative
   Frequency Table.

1. To explain how to calculate quartiles from
Cumulative Frequency Table.
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Statistics
Quartiles from Cumulative Frequency Table
S5 Int2
Example 1 The frequency table shows the
length of phone calls ( in minutes) made from
an office in one day.
Time Freq. (f)





Cum. Freq.
2
1
2
3
2
5
5
3
10
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8
4
18
4
5
22
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Statistics
S5 Int2
Quartiles from Cumulative Frequency Table
We use a combination of quartiles from a
frequency table and the Cumulative Frequency
Column.
For a list of 22 numbers, 22 4 5 R2
5 number
5 number
5 number
5 number
Q2
1 No.
1 No.
Q1
Q3
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The quartiles fall in the gaps between Q1 the
6th number Q1 3 minutes
Q2 the 11th and 12th number Q2 4 minutes
Q3 the 17th number. Q3 4 minutes
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Statistics
Quartiles from Cumulative Frequency Table
S5 Int2
Example 2 A selection of schools were asked
how many 5th year sections they have. Opposite
is a table of the results. Calculate the
quartiles for the results.
No. Of Sections Freq. (f)





Cum. Freq.
3
4
3
5
5
8
8
6
16
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9
7
25
8
8
33
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Statistics
S5 Int2
Quartiles from Cumulative Frequency Table
We use a combination of quartiles from a
frequency table and the Cumulative Frequency
Column.
Example 2 For a list of 33 numbers, 33 4 8
R1
8 number
8 number
8 number
8 number
Q1
Q3
1 No.
Q2
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The quartiles fall in the gaps between Q1 the
8th and 9th numbers Q1 5.5
Q2 the 17th number Q2 7
Q3 the 25th ad 26th numbers. Q3 7.5
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Statistics
S5 Int2
Quartiles from Cumulative Frequency Table
Now try Exercise 2 Ch11 (page 163)
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Starter Questions
S5 Int2
2cm
3cm
29o
4cm
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A
C
70o
53o
8cm
B
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Quartiles fromCumulative FrequencyGraphs
S5 Int2
Learning Intention
Success Criteria

1. Know the terms quartiles.

1. To show how to estimate quartiles from
cumulative frequency graphs.
2. Estimate quartiles from cumulative frequency
graphs.
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Quartiles fromCumulative FrequencyGraphs
S5 Int2
Number of sockets Cumulative Frequency
10 2
20 9
30 24
40 34
50 39
60 40
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New Term Interquartile range Semi-interquart
ile range (Q3 Q1 )2 (36 - 21)2 7.5
Cumulative FrequencyGraphs
S5 Int2
Quartiles
40 4 10
Q3
Q3 36
Q2
Q2 27
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Q1
Q1 21
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Quartiles fromCumulative FrequencyGraphs
S5 Int2
Km travelled on 1 gallon (mpg) Cumulative Frequency
20 3
25 11
30 30
35 53
40 69
45 76
50 80
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New Term Interquartile range Semi-interquart
ile range (Q3 Q1 )2 (37 - 28)2 4.5
Cumulative FrequencyGraphs
Cumulative FrequencyGraphs
S5 Int2
Q3
37
Quartiles
80 4 20
Q2
32
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Q1
28
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Quartiles fromCumulative FrequencyGraphs
S5 Int2
Now try Exercise 3 Ch11 (page 166)
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Starter Questions
S5 Int2
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Standard Deviation
S5 Int2
Learning Intention
Success Criteria

1. Know the term Standard Deviation.

1. To explain the term and calculate the
Standard Deviation for a collection of data.

1. Calculate the Standard Deviation for a collection
   of data.

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Standard Deviation For a FULL set of Data
S5 Int2
The range measures spread. Unfortunately any big
change in either the largest value or smallest
score will mean a big change in the range, even
though only one number may have changed.
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The semi-interquartile range is less sensitive to
a single number changing but again it is only
really based on two of the score.
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Standard Deviation For a FULL set of Data
S5 Int2
A measure of spread which uses all the data is
the Standard Deviation The deviation of a
score is how much the score differs from the mean.
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Step 1 Find the mean 375 5 75
Step 2 Score - Mean
Step 5 Take the square root of step
4 v13.6 3.7 Standard Deviation is 3.7 (to
1d.p.)
Step 4 Mean square deviation 68 5 13.6
Standard Deviation For a FULL set of Data
Step 3 (Deviation)2
S5 Int2
Example 1 Find the standard deviation of these
five scores 70, 72, 75, 78, 80.
Score Deviation (Deviation)2
70
72
75
78
80
Totals 375
-5
25
-3
9
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0
0
3
9
5
25
0
68
30
Step 1 Find the mean 180 6 30
Step 5 Take the square root of step
4 v160.33 12.7 (to 1d.p.) Standard Deviation
is 12.70
Step 2 Score - Mean
Step 4 Mean square deviation 962 6 160.33
Step 3 (Deviation)2
Standard Deviation For a FULL set of Data
S5 Int2
Example 2 Find the standard deviation of these
six amounts of money 12, 18, 27, 36, 37,
50.
Score Deviation (Deviation)2
12
18
27
36
37
50
Totals 180
-18
324
-12
144
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-3
9
6
36
7
49
20
400
0
962
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Standard Deviation For a FULL set of Data
S5 Int2
When Standard Deviation is HIGH it means the data
values are spread out from the MEAN.
When Standard Deviation is LOW it means the data
values are close to the MEAN.
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Mean
Mean
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Standard Deviation
S5 Int2
Now try Exercise 4 Ch11 (page 169)
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Starter Questions
S5 Int2
Waist Sizes Frequency
28 7
30 12
32 23
34 14
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Standard Deviation For a Sample of Data
S5 Int2
Learning Intention
Success Criteria

1. Construct a table to calculate the Standard
   Deviation for a sample of data.

1. To show how to calculate the Standard
deviation for a sample of data.
2. Use the table of values to calculate Standard
Deviation of a sample of data.
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Standard Deviation For a Sample of Data
We will use this version because it is easier to
use in practice !
S5 Int2
In real life situations it is normal to work with
a sample of data ( survey / questionnaire ).
We can use two formulae to calculate the sample
deviation.
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s standard deviation
? The sum of
n number in sample
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Step 2 Square all the values and find the total
Q1a. Calculate the mean 592 8 74
Step 3 Use formula to calculate sample
deviation
Step 1 Sum all the values
Q1a. Calculate the sample deviation
Standard Deviation For a Sample of Data
S5 Int2
Example 1a Eight athletes have heart rates
70, 72, 73, 74, 75, 76, 76 and 76.
Heart rate (x) x2
70
72
73
74
75
76
76
76
Totals
4900
5184
5329
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5476
5625
5776
5776
5776
?x2 43842
?x 592
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Q1b(ii) Calculate the sample deviation
Q1b(i) Calculate the mean 720 8 90
Standard Deviation For a Sample of Data
S5 Int2
Example 1b Eight office staff train as
athletes. Their Pulse rates are 80, 81, 83, 90,
94, 96, 96 and 100 BPM
Heart rate (x) x2
80
81
83
90
94
96
96
100
Totals
6400
6561
6889
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8100
8836
9216
9216
10000
?x 720
?x2 65218
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Q1b(iii) Who are fitter the athletes or
staff. Compare means Athletes are fitter
Q1b(iv) What does the deviation tell us. Staff
data is more spread out.
Standard Deviation For a Sample of Data
S5 Int2
Athletes
Staff
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Standard Deviation For a Sample of Data
S5 Int2
Now try Ex 5 6 Ch11 (page 171)
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40
Starter Questions
S5 Int2
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33o
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Scatter Graphs
S5 Int2
Construction of Scatter Graphs
Learning Intention
Success Criteria

1. Construct and understand the Key-Points of a
   scattergraph.

1. To construct and interpret Scattergraphs.

2. Know the term positive and negative
correlation.
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This scattergraph shows the heights and weights
of a sevens football team
Scatter Graphs
Write down height and weight of each player.
Construction of Scatter Graph
S5 Int2
Bob
Tim
Joe
Sam
Gary
Dave
Jim
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Scatter Graphs
Construction of Scatter Graph
S5 Int2
When two quantities are strongly connected we say
there is a strong correlation between them.
Best fit line
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Best fit line
Strong positive correlation
Strong negative correlation
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Scatter Graphs
Construction of Scatter Graph
S5 Int2
Key steps to Drawing the best fitting straight
line to a scatter graph

• Plot scatter graph.
• Calculate mean for each variable and plot the
• coordinates on the scatter graph.
• 3. Draw best fitting line, making sure it goes
  through
• mean values.

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Draw in the best fit line
Find the mean for theAge and Prices values.
Mean Age 2.9 Mean Price 6000
Scatter Graphs
Construction of Scatter Graph
S5 Int2
Is there a correlation? If yes, what kind?
Strong negative correlation
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Scatter Graphs
Construction of Scatter Graph
S5 Int2
Key steps to Finding the equation of the
straight line.

• Pick any 2 points of graph ( pick easy ones to
  work with).
• Calculate the gradient using
• Find were the line crosses yaxis this is b.
• Write down equation in the form y ax b

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Crosses y-axis at 10
Scatter Graphs
S5 Int2
Pick points (0,10) and (3,6)
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y -1.33x 10
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Scatter Graphs
Construction of Scatter Graph
S5 Int2
Now try Exercise 7 Ch11 (page 175)
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Starter Questions
S5 Int2
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Probability
S5 Int2
Learning Intention
Success Criteria

1. Understand the probability line.

1. To understand probability in terms of the number
   line and calculate simple probabilities.

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1. Calculate simply probabilities.

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ProbabilityLikelihood Line
S5 Int2
Certain
Evens
Impossible
Not very likely
Very likely
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Seeing a butterfly In July
Winning the Lottery
School Holidays
Baby Born A Boy
Go back in time
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ProbabilityLikelihood Line
S5 Int2
Certain
Evens
Impossible
Not very likely
Very likely
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It will Snow in winter
Everyone getting 100 in test
Homework Every week
Toss a coin That land Heads
Going without Food for a year.
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Probability
S5 Int2
We can normally attach a value to the
probability of an event happening.
To work out a probability
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P(A)
Probability is ALWAYS in the range 0 to 1
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ProbabilityNumber Likelihood Line
S5 Int2
0.1
0.2
0.3
0.4
0.6
0.7
0.8
0.9
Certain
Evens
Impossible
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8
P
1
Q. What is the chance of picking a number between
1 8 ?
8
4
Q. What is the chance of picking a number that is
even ?
0.5
P(E)
8
Q. What is the chance of picking the number 1 ?
1
0.125
P(1)
8
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ProbabilityLikelihood Line
S5 Int2
52 cards in a pack of cards
0.1
0.2
0.3
0.4
0.6
0.7
0.8
0.9
Certain
Evens
Impossible
Not very likely
Very likely
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0.5
P (Red)
Q. What is the chance of picking a red card ?
52
13
Q. What is the chance of picking a diamond ?
0.25
P (D)
52
4
Q. What is the chance of picking ace ?
P (Ace)
0.08
52
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Probability
S5 Int2
Now try Ex 8 Ch11 (page 177)
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Starter Questions
S5 Int2
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Relative Frequencies
S5 Int2
Learning Intention
Success Criteria

1. Know the term relative frequency.

1. To understand the term relative frequency.

1. Calculate relative frequency from data given.

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Relative Frequencies
Relative Frequency always added up to 1
S5 Int2
Relative Frequency How often an event happens
compared to the total number of events.
Example Wine sold in a shop over one week
Country Frequency Relative Frequency
France 180
Italy 90
Spain 90
Total
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0.5
180 360
0.25
90 360
0.25
90 360
1
360
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Relative Frequencies
S5 Int2
Example Calculate the relative frequency for
boys and girls born in the Royal Infirmary
hospital in December 2007.
Relative Frequency adds up to 1
Boys Girls Total
Frequency 300 200
Relative Frequency
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500
0.6
0.4
1
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Relative Frequencies
S5 Int2
Now try Ex 9 Ch11 (page 179)
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Starter Questions
S5 Int2
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Probability from Relative Frequency
S5 Int2
Learning Intention
Success Criteria

1. Know the term relative frequency.

1. To understand the connection of probability and
   relative frequency.

1. Estimate probability from the relative frequency.

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Probability from Relative Frequency
When the sum of the frequencies is LARGE the
relative frequency is a good estimate of the
probability of an outcome
S5 Int2
Example 1
Three students carry out a survey to study left
handedness in a school. Results are given below
Number of Left - Hand Students Total Asked Relative Frequency
Sean 2 10
Karen 3 25
Daniel 20 200
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Probability from Relative Frequency
Whos results would you use as a estimate of the
probability of a house being alarmed ?
Megans
S5 Int2
Example 2
Three students carry out a survey to study how
many houses had an alarm system in a particular
area. Results are given below
What is the probability that a house is alarmed ?
0.4
Number of Alarmed Houses Total Asked Relative Frequency
Paul 7 10
Amy 12 20
Megan 40 100
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Probability from Relative Frequency
S5 Int2
Now try Ex 10 Ch11 Start at Q2 (page 181)
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