Statistics: Data Presentation and Interpretation
Data presentation and interpretation covers the methods used to summarise, display and describe a set of data, including measures of location and spread, correlation, and diagrams such as histograms and box plots. A Level Statistics also covers identifying outliers, coding data, and interpreting scatter diagrams and box plots in context.
Method
- Calculate the mean using (sum of the values) / n, and for grouped data use the midpoint of each class in place of the actual values.
- Calculate the standard deviation using sqrt[(sum of x^2)/n - mean^2], using the midpoint of each class for grouped data.
- To find the median and quartiles from a list, use their position in the ordered data; for grouped data, use linear interpolation within the class that contains the median or quartile.
- Identify outliers using a stated rule, e.g. more than 2 standard deviations from the mean, or more than 1.5 x IQR beyond the nearer quartile, and check each value against the calculated boundary.
- Use a scatter diagram to describe correlation as positive, negative or (approximately) zero, and comment on how an unusual point would affect the product moment correlation coefficient.
- When comparing two data sets (e.g. using box plots), compare a measure of location (mean or median) and a measure of spread (range or IQR), always writing the comparison in context.
Worked example
The times, in minutes, taken by 8 runners to complete a 5 km route are: 28, 31, 26, 35, 30, 27, 33, 29. (a) Calculate the mean time. (b) Calculate the standard deviation of these times. (c) A ninth runner joins with a time of 50 minutes, a clear outlier. Without recalculating, state and justify the effect on the mean and standard deviation.
- Sum of the 8 times = 28+31+26+35+30+27+33+29 = 239. Mean = 239/8 = 29.875 minutes.
- Sum of squares = 28^2+31^2+26^2+35^2+30^2+27^2+33^2+29^2 = 784+961+676+1225+900+729+1089+841 = 7205.
- Variance = 7205/8 - 29.875^2 = 900.625 - 892.515625 = 8.109375. Standard deviation = sqrt(8.109375) = 2.848 (3dp).
- Adding 50 (well above the current mean of 29.875) would increase the mean, since it pulls the average up.
- The standard deviation would also increase, since 50 is a long way from the mean and adds a large deviation, increasing the overall spread.
Practice questions
Type your answer and press Check to be marked straight away, or reveal the answer and mark yourself.
Q1Calculate the mean of the data set: 4, 7, 9, 12, 13.Show answer
Answer: 9 (sum = 45, divided by 5).
Q2The lower quartile of a data set is 12 and the upper quartile is 20. Calculate the interquartile range.Show answer
Answer: 8 (20 - 12).
Q3In a scatter diagram, taller people tend to have larger shoe sizes. State the type of correlation shown between height and shoe size.Show answer
Answer: Positive correlation.
Q4For 6 values, sum of x = 90 and sum of x^2 = 1400. Calculate the standard deviation.Show answer
Answer: awrt 2.89 (sqrt(1400/6 - 15^2) = sqrt(8.33)).
Q5A data set has median 40, Q1 = 30 and Q3 = 52. Determine whether the value 90 is an outlier, using the rule that a value is an outlier if it lies more than 1.5 x IQR beyond the nearer quartile.Show answer
Answer: Yes; IQR = 22, so the upper boundary is 52 + 1.5(22) = 85, and 90 > 85.
Q6The heights, in cm, of 40 tomato plants are grouped: 20-30 (6 plants), 30-40 (14 plants), 40-50 (15 plants), 50-60 (5 plants). Using the midpoint of each class, calculate an estimate of the mean height.Show answer
Answer: 39.75 cm (sum of frequency x midpoint = 1590, divided by 40).
Exam-style questions
Written in the style of a A Level Maths exam paper, with a full mark scheme.
The mass, in kg, of 10 sacks of potatoes are: 24, 26, 22, 29, 25, 23, 28, 27, 26, 30. (a) Calculate the mean mass. (2) (b) Calculate the standard deviation of the masses. (2)
Show mark scheme
Tick each line you got. Your score builds from the marks on the scheme.
Nothing ticked yet - 4 available
The weekly hours worked by 11 part-time staff, listed in ascending order, are: 10, 12, 14, 15, 17, 18, 19, 21, 23, 25, 48. (a) Find the median, the lower quartile (Q1) and the upper quartile (Q3). (3) (b) Show that 48 hours is an outlier, using the rule that a value is an outlier if it lies more than 1.5 x IQR beyond the nearer quartile. (2)
Show mark scheme
Tick each line you got. Your score builds from the marks on the scheme.
Nothing ticked yet - 5 available
Two classes, C and D, sat the same physics test out of 60 marks. Box plots of their results gave the following five-figure summaries. Class C: minimum = 15, Q1 = 24, median = 33, Q3 = 44, maximum = 52. Class D: minimum = 20, Q1 = 30, median = 34, Q3 = 37, maximum = 40. (a) Compare the medians and the interquartile ranges of the two classes, in the context of the test. (3) (b) Using the quartiles, comment on the skewness of each class's distribution of marks. (3)
Show mark scheme
Tick each line you got. Your score builds from the marks on the scheme.
Nothing ticked yet - 6 available
Free printable worksheet
Want more practice on paper? Download the statistics: data presentation and interpretation worksheet pack - 11 pages of exam-style questions with a full mark scheme. One email opens every download in this browser for 14 days - no account, no card. Print it for personal and classroom use.
Next topics
Not quite what you needed?
Tell us what is missing on statistics: data presentation and interpretation, or which topic to write up next. Every request is read, and we reply to every one.
Build a full practice pack.
This topic is one of hundreds in the library - pick the ones a student needs and generate a printable PDF in minutes.