Amesa 2009. Jackie's workshop

1. The Interpretation of Quartiles
and Percentiles
Jackie Scheiber
RADMASTE Centre
Wits University
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2. Curriculum References
Grade 10 Grade 11 Grade 12
10.4.3 11.4.3 12.4.3
Understand that data Understand that data Understand that data can be
can be summarised in can be summarised summarised and compared in
different ways by and compared in different ways by calculating and
calculating and using different ways by using measures of central tendency
appropriate measures calculating and using and spread (distribution), inclusive
of central tendency measures of central of the
and spread tendency and spread •Mean
(distribution) to make (distribution) for more •Median
comparisons and draw than one set of data,
•Mode
conclusions, inclusive inclusive of the
of •Mean •Quartiles (Interpretation only)
•Mean •Median •Percentiles (interpretation only)
•Median •Mode
•Mode •Range
•Range
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3. Measures of Central Tendency
• A measure of central tendency is a
single number that can be used to
represent a set of data.
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4. • The three measures of central tendency
that are used are the MEAN, MEDIAN
and MODE
• The mean is the average found by
SHARING OUT EQUALLY the total of all
the values.
• The median is the MIDDLE VALUE when
all the values are placed in order of size.
• The mode is the value that occurs MOST
OFTEN.
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5. Which measure of central
tendency is best?
…It all depends …
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6. • The mean is a good summary for values that
represent magnitudes, like test marks and the
cost of something.
• The median is best used when ranking
people or things, like heights or when
extreme values might affect the mean.
• The mode is best used when finding out the
most popular dress size or the most popular
brand of chocolate.
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7. Measures of Central Tendency
and Measures of Spread or
Dispersion
• A measure of central tendency gives
you one data item that represents a set
of data.
• A measure of spread or dispersion
tells you how spread out the data items
are.
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8. Measures of Spread/Dispersion
• The range is the simplest measure of
spread. It is the difference between the
largest and the smallest values in the
data.
Range = largest value – smallest value
• This measure of spread does not take
into account anything about the
distribution of the data other than the
extremes.
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9. Another Measure of Dispersion
• A more trustworthy measure of spread
or dispersion is the range of the middle
half of the data.
• This measure of spread is the
Interquartile Range and is the
difference between the upper and the
lower quartiles. The Interquartile Range
(IQR) is not in the Maths Lit syllabus.
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10. QUARTILES
• A quartile divides a sorted data set
into 4 equal parts, so that each part
represents ¼ of the data set
Lower Median Upper
Quartile M Quartile
Q1
Q3
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11. Lower Quartile Median Upper Quartile
Q1 M Q3
• 25% of all the data has a value ≤ Q1
• 50% of all the data has a value ≤ M
• 75% of all the data has a value ≤ Q2
• 50% of all the data lies between
Q1 and Q3
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12. Lower
Median Upper
Quartile Quartile
M
Q1 Q3
• If a measurement falls to the right of the
upper quartile of a set of data, then we know
that it is in the top 25% of the data.
• We also know that it is better than at least 75% of
the data.
• If a measurement falls to the left of the lower
quartile of a set of data, then we know that it
is in the bottom 25% of the data.
• We also know that it is worse than at least 75% of
the data.
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13. Examples
• 3; 4; 5; 6; 6; 7; 8; 9; 9; 10; 11
There are 11 data items
The median is the 5th item. So M = 7
The lower quartile is the 3rd item. So Q1 = 5
The upper quartile is the 9th item. So Q3 = 9
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14. What does this mean?
3; 4; 5; 6; 6; 7; 8; 9; 9; 10; 11
Q1 M Q3
• ¼ or 25% of the data has a value that is less than or
equal to 5.
• ½ or 50% of the data has a value that is less than or
equal to 7
• ¾ or 75% of the data has a value that is less than or
equal to 9
• ½ or 50% of the data lies between 5 and 9
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15. Try the Activity on Page 190 & 191
1. Comment on Heights
• TALLEST
• 25% of the girls’ heights are between 150 cm and 160 cm
• 25% of the boys’ heights are between 160 cm and 170 cm
• CONCLUSION: all the boys in the top 25% are taller than
the girls
• MIDDLE 50%
• 50% of the girls have heights that are less than 140 cm
• 50% of the boys have heights that are less than 148 cm.
• The middle 50% of the girls have heights between 130 cm
and 150 cm
• The middle 50% of the boys have heights between 145
cm and 160 cm.
• CONCLUSION: In general the boys are taller than the
girls.
4. How many boys are taller than the tallest girl?
• 25% of the boys are taller than the tallest girl. 15

16. PERCENTILES
• A percentile is any of 99 values which divide
a sorted data set into 100 equal parts, so that
each part represents 1/100 of the data set
• If 70% of the population is shorter than you,
then your height is said to be at the 70th
percentile.
• The word percentile comes from the Latin
words per centum which means “per
hundred”
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17. • Percentiles are generally used with large sets
of data so that dividing it up into 100 equal
parts seems realistic.
• Suppose a test mark is calculated to be at the
84th percentile,
• then we know that 84% of the people who wrote
the test got the same mark or less than the test
mark
• We know that 16% of the people who wrote the
test scored higher than the test mark.
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18. • Sometimes
• Low percentile = good
• High percentile = good
It depends on the context …
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19. Example:
• A learner is given a test back. They got
a mark of 33. Is this a good mark or a
bad mark?
• Not sure
• If out of 35, is a good mark
• If out of 100, is a bad mark
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20. • Suppose we know this mark is at the
98th percentile. Is the mark good or
bad?
• It means that the learner did better than
98% of the rest of the learners.
• Is this good or bad?
• Suppose the mark is at the 3rd
percentile. Is this good or bad?
• It means that the learner did better than
3% of the rest of the learners.
• Is this good or bad?
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21. Examples:
1. Time taken to finish a test = 35 minutes.
This time was the lower quartile. What does
this mean?
xxxxx Q1 xxxxx M xxxxx Q3 xxxxx
35 min
• 25% of the learners finished the exam in 35
minutes or less
• 75% of the learners finished the exam in more
than 35 minutes
• Here a low quartile or percentile would be
considered good as finished more quickly on a
timed test is desirable. If take too long, won’t finish.
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22. • 70th percentile for a test was 16/20.
What does this mean?
1/20; 2/20; 5/20; 6/20; 12/20; 13/20; 16/20; 17/20..
smallest percentile  largest percentile
• 70% got 16/20 or less
• 30% got more than 16/20.
• Here a high percentile would be considered
good as answering more questions correctly
is desirable.
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23. Try the Activity on p 194 & 195
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24. 1) Runners in a race – want to finish the
race in a time that is less than
everyone else
• Low percentile is better – want fewer
people to have a time that is less than
yours.
• 20th percentile = 5,2 minutes.
• 20% of the people had a time that was
quicker/less than 5,2 minutes.
• 80% of the people had a time that was
slower/more than 5,2 minutes.
• This means that 5,2 minutes was a good
time.
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25. 1) Cyclists also want to finish the race is
LESS time than everyone else.
• 90th percentile = 1 hour 12 minutes. This
means that 90% of the cyclists finished in
1 hour 12 minutes or less. He is amongst
the slower cyclists in the race.
• 90% of the cyclists finished in 1 hour 12
minutes or less. 10% of the cyclists
finished in more than 1 hour 12 minutes.
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26. 1) For runners in a race, a higher speed
means a faster run.
• So the runners want a HIGH percentile
• 40th percentile = 12 km/h. This means
that 40% of the runners ran at LESS than
or equals to 12 km/h and 60% of the
runners ran at MORE than 12 km/h.
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27. 1. Exam – high or low percentile?
You want a mark that is better than the majority
of the learners – so you want a high percentile.
e.g. you want 90% of the learners to get a mark
that is less than or equal to yours.
• Waiting time of 32 min is in the upper
quartile. The less time you spend there, the
better
a) BAD
b) 75% of the people there waited for 32 minutes or
less than 32 minutes and 25% of the people
waited for more than 32 minutes
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28. • Mary’s salary was in the 78th
percentile.
• This means that
• 78% of the teachers got a salary that was
less than or equal to hers
• 22% of the teachers got a salary that was
more than hers.
• She should be pleased with the result.
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29. References
• Bloom Roberta: Descriptive Statistics:
Practice 3: Interpreting Percentiles
http://cnx.org/content/m18845/latest/
• Rolf HL: Finite Mathematics (2002)
Thomson Learning
• Tapson F: The Oxford Mathematics
Study Dictionary (2006) Oxford
University Press
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