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Numerical and Statistical Skills » Percentages, Trends and Misleading Data

What you'll learn this session

Study time: 30 minutes

AQA spec: 3.4.4

Calculate a percentage increase or decrease
Understand what a percentile tells you
Describe relationships in scatter plots, draw trend lines and make predictions
Spot when data has been presented in a misleading way

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Percentage increase and decrease

Percentages show change, but some data can mislead

A percentage change tells you how big a change is compared with where you started. It is fairer than just giving the difference, because a rise of 500 people means much more in a village than in a city.

Key terms:

Percentage change: the change in a value shown as a percentage of the original value.
Percentile: a value that has a given percentage of the data below it.
Bivariate data: two linked sets of data, shown on a scatter plot.
Trend line: a line that shows the general pattern in data.
Line of best fit: a straight line drawn through a scatter plot with about the same number of points on each side.
Interpolation: reading a value from inside the range of the data.
Extrapolation: extending a trend beyond the range of the data.

Use this method:

Find the change: new value minus original value.
Divide the change by the original value.
Multiply by 100.

If the answer is positive it is a percentage increase. If it is negative it is a percentage decrease.

Worked examples

Worked example: increase

A town had 8,000 residents in 2010 and 10,000 in 2020. Change = 10,000 - 8,000 = 2,000. 2,000 ÷ 8,000 = 0.25. 0.25 × 100 = 25% increase.

Worked example: decrease

A beach had 400 visitors a day in July and 300 a day in September. Change = 300 - 400 = -100. -100 ÷ 400 = -0.25. So there was a 25% decrease.

Common mistakes

Dividing by the new value instead of the original value. Forgetting to multiply by 100. Giving the raw change (2,000) when the question asks for a percentage.

Percentiles

Percentiles split ranked data into 100 equal parts. If a value is at the 90th percentile, then 90% of the data is below it and only 10% is above it.

📊 Reading a percentile

A pebble at the 80th percentile for length is longer than 80% of the pebbles measured.

🔗 Link to quartiles

The lower quartile is the 25th percentile, the median is the 50th and the upper quartile is the 75th.

You can find a percentile from a cumulative frequency graph. For the 90th percentile, work out 90% of the total frequency, find that value on the vertical axis, go across to the curve and then down to read the value on the horizontal axis.

Relationships in bivariate data

Bivariate data means two sets of data that are linked, such as distance from the sea and temperature. We show it on a scatter plot with one variable on each axis. Each point is one measurement.

Positive: as one variable rises, the other rises too.
Negative: as one variable rises, the other falls.
No relationship: the points are scattered with no pattern.

Say how strong it is too. Points close to the line show a strong relationship. Points spread widely show a weak one. Remember to say whether any anomalies sit far from the pattern.

Trend lines and lines of best fit

A line of best fit shows the trend in scattered data

A trend line can be sketched freehand through a scatter plot to show the general pattern. A line of best fit is an estimated straight line drawn with a ruler. Aim for roughly equal numbers of points above and below it. It does not have to go through every point, and it does not always have to go through the origin.

Worked example: prediction

A scatter plot shows pebble size falling as you move downstream. Your line of best fit passes through 90 mm at 2 km and 50 mm at 10 km. To estimate the size at 6 km, go up from 6 km on the horizontal axis to the line, then across to the vertical axis. The line passes through about 70 mm, so the estimate is about 70 mm.

Interpolation is a prediction inside the range you measured. It is usually reliable.
Extrapolation carries the line past your data. It is less reliable, because the pattern may change outside the range you measured. For example, pebbles cannot get smaller than zero.

Weaknesses in selective presentation of data

Data can be shown in a way that hides part of the picture. This is selective presentation. When you evaluate data, ask who made it and what might be missing.

Cherry-picking

Using only the results that support one view, such as only measuring on the day it was sunny.

Odd scales

A vertical axis that starts above zero makes a small change look huge.

Small samples

Results from only a few people or sites may not represent everyone.

Missing data: leaving out some years or places changes the story.
Percentage without the total: "a 50% rise" could be from 2 to 3 or from 2,000 to 3,000.
Choosing the time period: starting a graph at an unusual year can hide a longer pattern.

Common mistakes

Saying a line of best fit proves one thing causes another. A relationship shows a link, not proof of cause. Another factor could be behind both.

Exam-style question

A student counted visitors to a country park each year. In 2018 there were 20,000 visitors. In 2023 there were 26,000.

(a) Calculate the percentage increase in visitors from 2018 to 2023. [2 marks]

(b) The student drew a scatter graph of visitor numbers against distance from the car park for 12 points on a path. The points show a clear pattern of falling numbers. Describe the relationship. [2 marks]

(c) The student used the line of best fit to predict visitor numbers 3 km beyond the end of the data. Give one reason why this prediction may be unreliable. [1 mark]

Model answer

(a) Change = 26,000 - 20,000 = 6,000 (1). 6,000 ÷ 20,000 × 100 = 30% increase (1).
(b) A negative relationship (1): as distance from the car park increases, the number of visitors falls (1).
(c) Extrapolation goes beyond the measured data, so the pattern may not continue (1).

Exam tip

In (a), always divide by the original figure (2018), not the new one. Show each step so you can pick up the method marks.

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