Interpolation and extrapolation

Data measured over time are shown on a line chart: the time along the bottom axis, and each measurement a point.

Gal’s height, measured by the paediatrician every two years:

Only the points are measurements. The lines between them show the trend – how the height changed – but nobody measured Gal at age 7.

Interpolation is estimating a value that was not measured, between two data points.

How tall was Gal at age 7?

• Age 7 is exactly halfway between 6 and 8.
• At 6 the height was 115 cm, and at 8, 128 cm.
• An estimate: about halfway, $\displaystyle {\frac{115+128}{2}}={121.5}$ cm.

This is an estimate, not a measurement: perhaps Gal grew more in the first of the two years. But it is close, since there are measurements on both sides.

Extrapolation is estimating a value outside the span of the data: before the first data point or after the last. It is less reliable than interpolation, since outside the data the trend can change.

• Between ages 10 and 12 Gal grew 11 cm. If he goes on like this, at 14 he will be about $\displaystyle {149+11}={160}$ cm tall – a reasonable estimate, close to the data.
• Following the same trend – 5.5 cm a year – at 30 he would be about 250 cm tall. That will not happen: people stop growing at around 18.

The further from the data, the less reliable the estimate.

When the change is uniform – the same growth in every unit of time – the points lie on a straight line, and its formula can be written and used for estimates.

A seedling’s height, in cm, measured every two days:

• Every two days the seedling grows 3 cm, that is 1.5 cm a day: the slope $\displaystyle {m}={1.5}$.
• On day 0 the height is 4 cm: $\displaystyle {b}={4}$.
• The formula: $\displaystyle {y}={1.5x+4}$, where $\displaystyle x$ is the number of days and $\displaystyle y$ the height in cm.

• Interpolation, on day 3: $\displaystyle {y}={1.5\cdot3+4}={8.5}$ cm.
• Extrapolation, on day 10: $\displaystyle {y}={1.5\cdot10+4}={19}$ cm – an estimate that holds only if the seedling keeps growing at the same rate.

A town’s population, in thousands, was counted every five years.
The number for 2010 was lost:

Estimate how many people lived in the town in 2010. How did you estimate it?

Estimate how many people will live in the town in 2025.
Which of the estimates – for 2010 or for 2025 – is an interpolation, and which an extrapolation?

Omer carried the trend on to the year 2100, and got 160 thousand people.
Explain why his estimate is less reliable than the estimate for 2025.

A tank that holds 120 litres is being filled with water.
The water in the tank, in litres, every minute:

The points match the table. Write the formula of the trend: $\displaystyle y$ – the litres, $\displaystyle x$ – the minutes.

Estimate how much water was in the tank after two and a half minutes.

After how many minutes will the tank hold 110 litres? Write an equation and solve it.

By the formula, after 10 minutes the tank will hold 170 litres.
Explain why this cannot happen.

Noa’s phone is fully charged: 100%.
She is watching a film, and checks the battery every hour:

Write the formula of the trend: $\displaystyle y$ – the battery percentage, $\displaystyle x$ – the hours.

The phone warns when 10% are left. After how many hours will that happen, if the trend goes on? Write an equation and solve it.

Put $\displaystyle {x}={8}$ into the formula. What comes out, and what does it say about extrapolation?

A baby girl weighed 3.5 kg at birth, and 7 kg at six months.
Danny said: “She put on 3.5 kg in half a year. If she goes on like this, at 10 she will weigh 73.5 kg.”

Is Danny’s estimate reasonable? Explain.