Sec 4 E-Math · Statistics

Median, quartiles and percentiles questions with answers

Practice questions on median, quartiles and percentiles, from the basics up to exam level, each with a full worked solution. Try them on paper first, then open the solution to check your working.

Worked example

Find the median of $28, \ 41, \ 25, \ 37, \ 30, \ 45, \ 29$ .

Step 1 of 2

Put the data in order, smallest first, and count the values

25, \ 28, \ 29, \ 30, \ 37, \ 41, \ 45

n = 7

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Questions

Question 1Median of a list
Find the median of $34, \ 28, \ 41, \ 25, \ 37, \ 30, \ 45, \ 29$ .
Show the worked solution
1. Step 1: Put the data in order, smallest first, and count the values $25, \ 28, \ 29, \ 30, \ 34, \ 37, \ 41, \ 45$ $n = 8$
2. Step 2: Where is the median? It is the (n + 1) ÷ 2 th valuePosition: the 4.5th value, halfway between the 4th and 5th values. $\mathstrut \text{Median} = \dfrac{30 + 34}{2}$ $\text{Median} = 32$
Question 2Quartiles of a list
Find the upper quartile of $48, \ 52, \ 39, \ 61, \ 44, \ 57, \ 50, \ 46, \ 55, \ 42$ .
Show the worked solution
1. Step 1: Put the data in order, smallest first, and count the values $39, \ 42, \ 44, \ 46, \ 48, \ 50, \ 52, \ 55, \ 57, \ 61$ $n = 10$
2. Step 2: Split into a lower and an upper half $\text{Lower half: } 39, \ 42, \ 44, \ 46, \ 48$ $\text{Upper half: } 50, \ 52, \ 55, \ 57, \ 61$
3. Step 3: The quartiles are the medians of the halves $Q_1 = 44$ $Q_3 = 55$
Question 3From a cumulative frequency curve
The cumulative frequency curve shows the marks of $50$ students. Use the curve to estimate the 80th percentile.
Show the worked solution
1. Step 1: Which cumulative frequency do you go across from? $\mathstrut \dfrac{80}{100} \times 50 = 40$
2. Step 2: Go across from that cumulative frequency to the curve, then down to the axis $P_{80} \approx 40.0$
Question 4Percentile of a mark; a mark from a probability
The cumulative frequency curve shows the marks of $80$ students. The probability that a student chosen at random scored more than $t$ marks is $0.3$ . Use the curve to estimate $t$ .
Show the worked solution
1. Step 1: Turn the probability into a number of studentsMore than $t$ marks: $0.3 \times 80 = 24$ students
2. Step 2: The curve counts the students with AT MOST a mark: take them from the totalAt most $t$ marks: $80 - 24 = 56$ students
3. Step 3: Go across from that cumulative frequency to the curve, then down $t \approx 38.1$

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Worked example

Find the median of

28, \ 41, \ 25, \ 37, \ 30, \ 45, \ 29

Step 1 of 2

Put the data in order, smallest first, and count the values

25, \ 28, \ 29, \ 30, \ 37, \ 41, \ 45

n = 7

Questions

Question 1Median of a list

Find the median of $34, \ 28, \ 41, \ 25, \ 37, \ 30, \ 45, \ 29$ .

Show the worked solution

Step 1: Put the data in order, smallest first, and count the values $25, \ 28, \ 29, \ 30, \ 34, \ 37, \ 41, \ 45$ $n = 8$
Step 2: Where is the median? It is the (n + 1) ÷ 2 th valuePosition: the 4.5th value, halfway between the 4th and 5th values. $\mathstrut \text{Median} = \dfrac{30 + 34}{2}$ $\text{Median} = 32$

Question 2Quartiles of a list

Find the upper quartile of $48, \ 52, \ 39, \ 61, \ 44, \ 57, \ 50, \ 46, \ 55, \ 42$ .

Show the worked solution

Step 1: Put the data in order, smallest first, and count the values $39, \ 42, \ 44, \ 46, \ 48, \ 50, \ 52, \ 55, \ 57, \ 61$ $n = 10$
Step 2: Split into a lower and an upper half $\text{Lower half: } 39, \ 42, \ 44, \ 46, \ 48$ $\text{Upper half: } 50, \ 52, \ 55, \ 57, \ 61$
Step 3: The quartiles are the medians of the halves $Q_1 = 44$ $Q_3 = 55$

Question 3From a cumulative frequency curve

The cumulative frequency curve shows the marks of $50$ students. Use the curve to estimate the 80th percentile.

Show the worked solution

Step 1: Which cumulative frequency do you go across from? $\mathstrut \dfrac{80}{100} \times 50 = 40$
Step 2: Go across from that cumulative frequency to the curve, then down to the axis $P_{80} \approx 40.0$

Question 4Percentile of a mark; a mark from a probability

The cumulative frequency curve shows the marks of $80$ students. The probability that a student chosen at random scored more than $t$ marks is $0.3$ . Use the curve to estimate $t$ .

Show the worked solution

Step 1: Turn the probability into a number of studentsMore than $t$ marks: $0.3 \times 80 = 24$ students
Step 2: The curve counts the students with AT MOST a mark: take them from the totalAt most $t$ marks: $80 - 24 = 56$ students
Step 3: Go across from that cumulative frequency to the curve, then down $t \approx 38.1$