The Interpretation of Quartiles and Percentiles

Jackie Scheiber

RADMASTE Centre

Wits University

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Curriculum References

Grade 10 Grade 11 Grade 1210.4.3

Understand that data can be summarised in different ways by calculating and using appropriate measures of central tendency and spread (distribution) to make comparisons and draw conclusions, inclusive of•Mean•Median•Mode•Range

11.4.3

Understand that data can be summarised and compared in different ways by calculating and using measures of central tendency and spread (distribution) for more than one set of data, inclusive of the•Mean•Median•Mode•Range

12.4.3

Understand that data can be summarised and compared in different ways by calculating and using measures of central tendency and spread (distribution), inclusive of the•Mean•Median•Mode•Quartiles (Interpretation only)•Percentiles (interpretation only)

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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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• 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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Which measure of central tendency is best?

…It all depends …

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• 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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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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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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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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QUARTILES

• A quartile divides a sorted data set into 4 equal parts, so that each part represents ¼ of the data set

Lower Quartile

Q1

MedianM

Upper Quartile

Q3

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• 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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Lower QuartileQ1

MedianM

Upper QuartileQ3

• 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.

Lower Quartile

Q1

MedianM

Upper Quartile

Q3

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Examples

• 3; 4; 5; 6; 6; 7; 8; 9; 9; 10; 11There 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 = 913

What does this mean?3; 4; 5; 6; 6; 7; 8; 9; 9; 10; 11

• 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

Q1M Q3

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Try the Activity on Page 190 & 1911. 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.

2. How many boys are taller than the tallest girl?• 25% of the boys are taller than the tallest girl. 15

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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• 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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• Sometimes• Low percentile = good• High percentile = good

It depends on the context …

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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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• 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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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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2. 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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Try the Activity on p 194 & 195

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1) Runners in a race – want to finish the race in a time that is less than everyone else

a) Low percentile is better – want fewer people to have a time that is less than yours.

b) 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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2) Cyclists also want to finish the race is LESS time than everyone else.

a) 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.

b) 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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3) For runners in a race, a higher speed means a faster run.

a) So the runners want a HIGH percentile

b) 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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4. 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.

5. 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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6. 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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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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