Year 10 Extended · Unit 6 Statistics and the bell curve · Lesson 5 of 7

What happens to the measures when every value changes

Convert every value to new units, or add the same bonus to all of them, and the measures of centre and spread respond in a pattern. Finding that pattern, then proving it from the formula, is the aim of this investigation.

Statistics and ProbabilityGDC optional

Warm-up

Three quick ones from earlier work. Try each in your head or on paper before you open the answer.

  1. From Lesson 3: a set has σ = 2.5. What is its variance?
    Answer6.25. The variance is σ², and 2.5 × 2.5 = 6.25.
  2. From Year 8: find the median of 3, 5, 8, 12 and 20.
    Answer8. It is the middle of five ordered values.
  3. From Year 9: expand 3(x − 2)².
    Answer3x² − 12x + 12. Square the bracket first, x² − 4x + 4, then multiply by 3.
Words used in this lesson
shift
add the same number to every value. For example: adding 4 to every length.
scale
multiply every value by the same number. For example: converting metres to centimetres.
transformation
a change applied to every value in the same way. For example: x → 2x − 7.
centre
a measure of where the data sit, such as the mean. For example: the median of 3.5.
spread
a measure of how scattered the data are. For example: the IQR of 0.8.
invariant
unchanged by a transformation. For example: the range under a shift.

Marks out of 20, marks out of 40

Seven pupils in a class in Accra scored 4, 6, 7, 9, 10, 12 and 15 on a quiz marked out of 20. The teacher may add the same bonus to every mark, or double every mark to put the quiz out of 40.

Predict first: for each change, which of the mean, median, range, IQR and standard deviation will change, and by how much? Write your predictions down, then use the figure to test them.

1
0
9mean
6interquartile range
3.46standard deviation

new = 1 × old + 0: mean 9, median 9, range 11, IQR 6, standard deviation 3.46.

Show me the steps
  1. Set k = 1 and c = 5. Every dot moves 5 to the right. The mean and the median become 14. The range 11, the IQR 6 and the standard deviation 3.46 do not change: the shape is the same, in a new place.
  2. Set c = 0 and k = 2. Every dot is twice as far from zero. The mean and median become 18, and every spread doubles: range 22, IQR 12, standard deviation 6.93.
  3. Set k = −2 with c = 0. The dots flip across zero, so the mean is −18. The spreads are still 22, 12 and 6.93: a spread is a distance, so only the size of k matters.
  4. Set k = 2 and c = 5. The mean is 2 × 9 + 5 = 23, and the standard deviation is twice the original, 6.93. The 5 appears in the centre and never in the spread.

If every value x becomes kx + c:

  • the mean and median follow the same rule: new mean = k × old mean + c;
  • the range, IQR and standard deviation are multiplied by |k|, the size of k, and c plays no part.

Why. The deviation of kx + c from its new mean kμ + c is k(x − μ), so every deviation is multiplied by k and every squared deviation by k². Hence σ′² = k²σ², and σ′ = |k|σ.

Adding does not change the spread. The sets 3, 6, 9 and 23, 26, 29 have different means but the same standard deviation, because every deviation from the mean is the same. A spread is a distance, and moving a whole data set along the number line does not change any distance in it.

Worked examples

1. The data 2, 5, 8 become 3x − 1. Find the new mean and standard deviation.

  1. Old: mean 5, deviations −3, 0, 3, variance 18 ÷ 3 = 6, σ = √6 = 2.45.
  2. New values are 5, 14, 23. The mean is 14, which is 3 × 5 − 1.
  3. The new deviations are −9, 0, 9, with variance 162 ÷ 3 = 54 and σ = √54 = 7.35, which is 3 × 2.45.

2. Daily mean temperatures in a Kenyan town have mean 14.2°C and σ 3.1°C. Convert to kelvin, K = C + 273.

  1. Here k = 1 and c = 273. The mean is 14.2 + 273 = 287.2 K.
  2. The standard deviation is 1 × 3.1 = 3.1 K, unchanged.

3. Prove that the standard deviation of kx + c is |k| times that of x.

  1. The new mean is kμ + c, so each deviation is (kx + c) − (kμ + c) = k(x − μ).
  2. σ′² = Σk²(x − μ)² ÷ n = k²σ², so σ′ = √(k²σ²) = |k|σ.

Your go. The first step is given. Heights in a class have mean 1.64 m and standard deviation 0.08 m. Find the mean and standard deviation in centimetres.

  1. Converting to centimetres multiplies every height by 100, so k = 100 and c = 0.
  2. Now apply the rule to the mean and to the standard deviation.
Check the answer

The mean is 100 × 1.64 = 164 cm and the standard deviation is 100 × 0.08 = 8 cm. Nothing was added, so c plays no part.

With a GDC (optional)

Put the data in one list, then define a second list as k times the first plus c. Ask for the one-variable statistics of both lists and compare the mean, median, quartiles and standard deviation side by side.

The display confirms a pattern; it does not prove it. The proof comes from the formula for σ. Everything here can be done by hand.

Check before you go on

1. Every value in a data set is multiplied by 4 and then 3 is added. The standard deviation was 5. What is the new standard deviation?

Your turn

Start at Level 1 and go as far as you can. Type negative numbers with a minus sign, for example -7. A wrong answer gets a reason, not just a cross: read it, then try again.

Level 1 · Getting started

2. A data set has mean 12.5, median 11 and standard deviation 2.4. Every value is increased by 8. What is the new median?

3. Lengths in centimetres have standard deviation 4.5. Every length is multiplied by 10 to give millimetres. What is the new standard deviation?

Level 2 · Building

4. Daily temperatures in degrees Celsius have mean 22 and standard deviation 3. They are converted by F = 1.8C + 32. What is the standard deviation in degrees Fahrenheit, to one decimal place?

5. A data set has mean 8. Every value is multiplied by −2 and then 10 is added. What is the new mean?

Level 3 · Stretch

6. Every value is changed by y = 3x − 4. The new data have mean 20 and standard deviation 12. What was the standard deviation of the original data?

7. A set of 10 values has Σ(x − μ)² = 50. Every value is multiplied by −3. What is the new standard deviation, to one decimal place?

Something to think about

Lengths in centimetres are converted to inches by dividing every value by 2.54. Which of the mean, the median, the range and the standard deviation are divided by 2.54, and is there a measure here that stays the same? What would stay the same if a constant were added instead?

One way to think about it

All four are divided by 2.54, because dividing every value by 2.54 scales every position and every distance by the same factor, 1 ÷ 2.54. None of them stays the same. Adding a constant is what leaves the range and the standard deviation unchanged, since it slides every value and moves no distance.

Exit ticket

8. A data set has mean 50 and IQR 8. Every value x is changed to 20 − 0.5x. Which gives the new mean and IQR?