Scatter Graphs and Correlation
A scatter graph plots pairs of related data as points on a grid, one axis for each variable, to show whether a correlation exists between them: positive correlation means both variables increase together, negative correlation means one increases as the other decreases, and no correlation means there is no clear pattern. A line of best fit, a straight line drawn through the data, estimates values, more reliably within the data's range, interpolation, than beyond it, extrapolation, and an outlier is a point that clearly does not fit the pattern.
Before you start
Make sure you're comfortable with these topics first:
Method
- Plot each pair of values as a point, reading both axes' scales carefully - they often do not start at 0 or increase in equal steps of 1.
- Look at the overall pattern of the points to decide the type of correlation: positive, negative or none.
- Describe the strength as well as the type: points lying close to a line show strong correlation; points scattered widely around a rough line show weak correlation.
- Identify any outlier - a point that clearly does not fit the pattern of the rest of the data.
- If asked, draw a line of best fit by eye, with roughly equal numbers of points above and below it, and ignore any outlier when doing so.
- To estimate a value, use the line of best fit or the trend between the nearest data points, and state whether the estimate is interpolation (inside the data's range, more reliable) or extrapolation (outside it, less reliable).
Worked example
The table shows the temperature, in degrees Celsius, and the number of hot chocolates sold at a cafe on 6 days: Temperature: 2, 5, 8, 11, 14, 17. Hot chocolates sold: 48, 42, 33, 25, 20, 12. Describe the correlation, and use the trend to estimate the number of hot chocolates sold on a day with a temperature of 10 degrees.
- As the temperature increases (2 up to 17), the number of hot chocolates sold decreases (48 down to 12), so this is negative correlation.
- The values fall fairly steadily as temperature rises, so this is a strong negative correlation.
- 10 degrees lies between the given values of 8 (33 sold) and 11 (25 sold), so estimating here is interpolation, which is reasonably reliable.
- Between 8 and 11 degrees, sales fall by 33 - 25 = 8 over a rise of 3 degrees, a rate of about 2.7 fewer sold per degree.
- From 8 to 10 degrees is a rise of 2 degrees, so estimate the fall as about 2 x 2.7 = 5.3, giving 33 - 5.3, approximately 28 hot chocolates sold at 10 degrees. On an exam grid you would draw a line of best fit through all six points and read off the value at 10 degrees instead; any sensible answer close to the trend would be accepted.
Practice questions
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Q1A student records the number of hours of revision and the test score for 5 pupils: (1 hour, 40 marks), (2 hours, 52 marks), (3 hours, 58 marks), (4 hours, 65 marks), (5 hours, 80 marks). What type of correlation does this data show?Show answer
Answer: Positive correlation (as hours of revision increases, test score increases)
Q2A scatter graph plots the age of 10 cars against their resale value. As the cars get older, their resale value tends to fall. What type of correlation is this?Show answer
Answer: Negative correlation
Q3A scatter graph plots pupils' shoe size against their score in a spelling test, and there is no clear pattern between the two. What type of correlation, if any, does this show?Show answer
Answer: No correlation (shoe size has no relationship with spelling ability)
Q4Six pupils' maths and science test scores, out of 50, are: (30, 32), (35, 38), (40, 41), (45, 44), (48, 47), (12, 45). Which pair of results is an outlier, and why?Show answer
Answer: (12, 45) is the outlier: the other five pupils all have similarly matched maths and science scores, but this pupil's very low maths score of 12 does not fit that pattern alongside a high science score of 45
Q5A line of best fit is drawn through data with x-values between 5 and 20. State whether using this line to estimate y when x = 12 is interpolation or extrapolation, and whether it is likely to be reliable.Show answer
Answer: Interpolation, since x = 12 lies within the range of the data (5 to 20); this is generally reliable
Q6Using the same line of best fit, drawn for x-values between 5 and 20, a student uses it to estimate y when x = 50. State whether this is interpolation or extrapolation, and comment on its reliability.Show answer
Answer: Extrapolation, since x = 50 is well outside the range of the data (5 to 20); this estimate is much less reliable, as the trend might not continue beyond the data collected
Q7Two scatter graphs both show positive correlation. In graph A, the points lie very close to a straight line. In graph B, the points are more spread out around a rough upward trend. Which graph shows the stronger correlation?Show answer
Answer: Graph A, since points lying closer to the line indicate a stronger correlation
Q8A line of best fit passes through the points (0, 10) and (20, 50) on a scatter graph of x against y. Assuming y increases at a constant rate along the line, estimate the value of y when x = 10.Show answer
Answer: y = 30 (x = 10 is halfway between 0 and 20, so y is halfway between 10 and 50)
Exam-style questions
Written in the style of a KS3 Maths exam paper, with a full mark scheme.
The table shows the number of ice creams sold and the daily maximum temperature at a seaside kiosk over 6 days: Temperature (C): 14, 17, 19, 22, 25, 28. Ice creams sold: 40, 55, 65, 90, 110, 130. (a) State the type of correlation shown by this data. (b) A line of best fit is used to estimate ice cream sales at 20 degrees C. State whether this involves interpolation or extrapolation. (c) Using the trend in the table, estimate the number of ice creams sold at 20 degrees C.
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A scientist records the number of hours of sunlight and the height growth, in mm, of 6 seedlings over a week: Hours of sunlight: 10, 15, 20, 25, 30, 35. Height growth (mm): 8, 11, 15, 19, 3, 27. (a) Identify which pair of results is an outlier. (b) Ignoring the outlier, describe the correlation shown by the rest of the data. (c) Explain why the outlier should usually be ignored when drawing a line of best fit.
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A researcher measures the height of a sunflower every week for the first 8 weeks after planting and finds a strong positive correlation between the number of weeks and the height of the plant. She uses a line of best fit to predict the height of the sunflower after 2 years (104 weeks). Explain two reasons why this prediction is unlikely to be reliable.
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Free printable worksheet
Want more practice on paper? Download the scatter graphs and correlation worksheet pack - 8 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.
This topic is chapter 35 of KS3 Maths Workbook 2, the whole course as one free printable PDF.
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