Descriptive statistics: practice problems with step-by-step solutions
Also called: mean, median and standard deviation
Free Descriptive statistics practice problems with diagrams, hints and full worked solutions. New values every time, at three levels: mean and median, standard deviation, normal distribution.
Free practice problemEasy
Find the mean x̄ of the data: 4, 14, 15, 20, 40, 37, 14, 33, 33.
Descriptive statistics summarise a list of numbers, called data, with a few values. The first question is: where is the centre?
The mean is the ordinary average: add all values and divide by how many there are, n.
x̄ = Σx / n
The symbol Σ means “add them all up”, and x̄ is read “x bar”. For 3, 5 and 10 the mean is 18/3 = 6.
The median is the middle value after you sort the data from small to large.
Odd n: the median is the single middle value. For 3, 5, 10 it is 5.
Even n: the median is the average of the two middle values.
One extreme value, an outlier, pulls the mean towards it but hardly moves the median. Incomes are often summarised by the median for that reason.
Self-check What is the median of 4, 40, 29, 22, 17, 20?
20
21
22
25.5
Answer: 21. Sorted: 4, 17, 20, 22, 29, 40. The two middle values are 20 and 22, so the median is 21. 25.5 comes from taking the middle of the unsorted list; 22 is the mean.
2The spread: standard deviation
Two data sets can have the same mean but a very different spread. The standard deviation s measures how far values typically lie from the mean.
Each value has a deviation x − x̄. Deviations add up to zero, so you square them first, which makes them all positive.
s = √( Σ(x − x̄)² / (n − 1) )
This is the sample standard deviation: you divide by n − 1, not by n. The square of s is called the variance.
For 2, 4 and 6: the mean is 4, the deviations are −2, 0 and 2, the squares add up to 8, 8/2 = 4, and s = √4 = 2.
Self-check What is the sample standard deviation of 2, 4 and 6?
0
1.633
2
4
Answer: 2. Σ(x − x̄)² = 8, divided by n − 1 = 2 gives 4, and √4 = 2. Dividing by n = 3 gives 1.633; 4 forgets the square root, and 0 is the sum of the deviations before squaring.
3z-scores and the normal distribution
Many measurements, such as heights or exam scores, follow a bell-shaped curve: the normal distribution. Its centre is the mean μ and its width is the standard deviation σ.
A z-score says how many standard deviations a value lies from the mean:
z = (x − μ) / σ
With μ = 70 and σ = 10, a score of 74 has z = 4/10 = 0.4. A negative z means the value lies below the mean.
The 68–95–99.7 rule: about 68% of values lie within 1σ of the mean, 95% within 2σ and 99.7% within 3σ.
With μ = 50 and σ = 5, the value 60 has z = 2. About 95% lies between 40 and 60, so about 2.5% lies above 60. A z-table gives the exact value 0.0228.
Self-check IQ scores are normal with μ = 100 and σ = 15. About what percentage of people score above 115?
16%
32%
34%
68%
Answer: 16%. 115 is 1σ above the mean. 68% lies within 1σ, so 32% lies outside, split equally over both tails: 16% above. 32% forgets to split the two tails.
4The method step by step
Write the data in a row and count n.
Mean: add everything and divide by n.
Median: sort first, then take the middle value, or the average of the two middle values.
Standard deviation: subtract the mean from each value, square, add, divide by n − 1, take the square root.
z-score: subtract μ, divide by σ. Then use the 68–95–99.7 rule or a z-table for a proportion.
A table with the columns x, x − x̄ and (x − x̄)² keeps the standard deviation calculation tidy.
5A worked example
Follow every step. Try to predict the next one before you open it.
Worked example
Find the mean, the median and the sample standard deviation s of the data: 4, 8, 6, 5, 7.
1
The mean
x̄ = (4 + 8 + 6 + 5 + 7) / 5 = 30 / 5 = 6
2
The median
Sorted: 4, 5, 6, 7, 8. With n = 5 the middle value is the third one.
median = 6
3
Deviations and squares
Subtract the mean from every value: −2, 2, 0, −1, 1. Check: they add up to 0.
Square them: 4, 4, 0, 1, 1.
Σ(x − x̄)² = 4 + 4 + 0 + 1 + 1 = 10
4
Divide by n − 1
s² = 10 / (5 − 1) = 2.5
5
Take the square root
s = √2.5 ≈ 1.58
Check: the values lie 0, 1 or 2 away from the mean, so a typical distance of about 1.6 makes sense.
Answer
Mean x̄ = 6, median 6, and s ≈ 1.58.
6Common mistakes
Taking the middle of the list without sorting it first.
Dividing by n instead of n − 1 for a sample standard deviation.
Forgetting the square root at the end, which gives the variance instead of s.
Getting the sign of a z-score wrong: a value below the mean has a negative z.
Forgetting to divide by σ in the z-score.
Self-check Exam scores have mean 70 and standard deviation 10. What is the z-score of a score of 60?
1
6
−10
−1
Answer: −1. z = (60 − 70)/10 = −10/10 = −1. The score lies below the mean, so z is negative. −10 forgets to divide by σ.
7How exam questions look
Easy: the mean or the median of a short list of data.
Medium: the sample standard deviation of five or six values. Divide by n − 1.
Hard: the normal distribution. Find a z-score, or the proportion of values above or below a given value.
A quick check: the mean and the median lie between the smallest and the largest value, and s is never negative.
Find the median M of the data: 39, 11, 3, 28, 18, 17, 40, 40, 5.
n = 9
Find M
Show hints
Sort the values from smallest to largest.
With an odd number of values the median is the middle one.
sorted: 3, 5, 11, 17, 18, 28, 39, 40, 40.
Show the answer and the step-by-step solution
Descriptive statistics: the mean is the sum divided by n; the median is the middle value of the sorted data (the average of the two middle values when n is even). The sample standard deviation is s = √(Σ(x − x̄)²/(n − 1)). A z-score z = (x − μ)/σ says how many standard deviations a value lies from the mean.
Sort
3, 5, 11, 17, 18, 28, 39, 40, 40
Middle value
M = 18
Answer
M = 18
Example 2Medium
Find the sample standard deviation s of the data: 26, 57, 34, 22, 53, 13, 59.
n = 7
Find s
Show hints
First find the mean.
Square each deviation from the mean, add them, divide by n − 1 and take the square root.
x̄ = 37.71.
Show the answer and the step-by-step solution
Descriptive statistics: the mean is the sum divided by n; the median is the middle value of the sorted data (the average of the two middle values when n is even). The sample standard deviation is s = √(Σ(x − x̄)²/(n − 1)). A z-score z = (x − μ)/σ says how many standard deviations a value lies from the mean.
A quantity is normally distributed with mean 60 and standard deviation 12. What proportion P of values is below 36?
μ = 60σ = 12x = 36
Find P
Show hints
Convert to a z-score first.
z = (36 − 60)/12 = −2.
Read Φ(z) from the standard normal table.
Show the answer and the step-by-step solution
Descriptive statistics: the mean is the sum divided by n; the median is the middle value of the sorted data (the average of the two middle values when n is even). The sample standard deviation is s = √(Σ(x − x̄)²/(n − 1)). A z-score z = (x − μ)/σ says how many standard deviations a value lies from the mean.
Standardise
z = −2
Standard normal table
Φ(−2) = 0.02275
P = 0.02275
Answer
P = 0.02275
Questions and answers
What is Descriptive statistics?
Descriptive statistics summarise a list of numbers, called data, with a few values. The first question is: where is the centre?
The mean is the ordinary average: add all values and divide by how many there are, n.
How do I solve Descriptive statistics problems step by step?
Descriptive statistics: the mean is the sum divided by n; the median is the middle value of the sorted data (the average of the two middle values when n is even). The sample standard deviation is s = √(Σ(x − x̄)²/(n − 1)). A z-score z = (x − μ)/σ says how many standard deviations a value lies from the mean.
What do the Descriptive statistics exercises cover?
Easy: mean and median
Medium: standard deviation
Hard: normal distribution
Every problem has new values, so you learn the method instead of an answer. Stuck? Open a small hint first, and the full explanation only when you need it.
How practising works
Every problem has new values, so you learn the method instead of an answer.
Stuck? Open a small hint first, and the full explanation only when you need it.
Your progress, mistakes and saved problems are kept, so you can review before the exam.