Changing every value Problem set B One question at a time

Find the new mean and standard deviation of each data set.

1. (a) The mean is 72 and the standard deviation is 6.4. Then 18 is added to every value.

   • Mean: 72 + 18 = 90
   • Standard deviation unchanged: 6.4

2. (b) The mean is 410 and the standard deviation is 27. Then 90 is subtracted from every value.

   • Mean: 410 − 90 = 320
   • Standard deviation unchanged: 27

Find the new median and IQR.

1. (a) A data set has median 91 and IQR 24. Every value is decreased by 40.

   • Median: 91 − 40 = 51
   • IQR unchanged: 24

2. (b) A data set has median 12.5 and IQR 3.2. Every value is increased by 4.5.

   • Median: 12.5 + 4.5 = 17
   • IQR unchanged: 3.2

Find the new median and IQR.

1. (a) A data set has median 9 and IQR 3.5. Every value is multiplied by 8.

   • Median: 9 × 8 = 72
   • IQR: 3.5 × 8 = 28

2. (b) A data set has median 260 and IQR 52. Every value is divided by 5.

   • Median: 260 ÷ 5 = 52
   • IQR: 52 ÷ 5 = 10.4

Find the new mean and standard deviation.

1. (a) The mean is 16 and the standard deviation is 2.5. Every value is multiplied by 4.

   • Mean: 16 × 4 = 64
   • SD: 2.5 × 4 = 10

2. (b) The mean is 36 and the standard deviation is 4.5. Every value is divided by 9.

   • Mean: 36 ÷ 9 = 4
   • SD: 4.5 ÷ 9 = 0.5

A data set has standard deviation 1.5. Every value is multiplied by 4. Find the new standard deviation and the new variance.

• New SD = 4 × 1.5 = 6
• New variance = 6² = 36
• Check: the old variance 2.25 is multiplied by 4² = 16: 36

The scores of a class on a quiz have mean 46, standard deviation 7.5, median 44 and IQR 9. The teacher adds 5 marks to every score. Write down the new mean, standard deviation, median and IQR.

• Mean: 46 + 5 = 51
• Median: 44 + 5 = 49
• SD and IQR measure spread, so they do not change: 7.5 and 9

The marks of a class have mean 52 and standard deviation 10. The marks are rescaled: new mark = 1.5 × old mark + 8. Find the mean and standard deviation of the new marks.

• Mean: 1.5 × 52 + 8 = 86
• SD: 1.5 × 10 = 15
• Adding 8 does not change the standard deviation

A café records how many cups of tea it sells in five hours: 9, 11, 14, 15, 21. Over a festival, sales in each hour are three times as many, plus 2 extra cups.

1. (a) Find the mean and standard deviation of the original data.

   • From the calculator: x̄ = 14, σx = 4.10

2. (b) Write down the new data, and find its mean and standard deviation from the calculator.

   • New data: 29, 35, 44, 47, 65
   • From the calculator: x̄ = 44, σx = 12.3

3. (c) Find the new mean and standard deviation from the original ones, without the new data, and check they agree.

   • Mean: 3 × 14 + 2 = 44
   • σ: 3 × 4.10 = 12.3
   • These agree with the calculator

Every value in a data set is multiplied by k (k > 0) and then b is added. The original mean is 20 and the original standard deviation is 4. The new mean is 47 and the new standard deviation is 8. Find k and b.

• New SD = k × 4 = 8, so k = 8 ÷ 4 = 2
• New mean = k × 20 + b = 47
• 2 × 20 + b = 47, so b = 47 − 40 = 7

The journey times of a bus route have mean 30 minutes and standard deviation 5 minutes. New traffic lights make every journey 10% shorter, and a new stop then adds 4 minutes to every journey. Find the mean and standard deviation of the new journey times.

• 10% shorter means multiplying by 0.9
• Mean: 0.9 × 30 + 4 = 31
• SD: 0.9 × 5 = 4.5 (adding 4 does not change it)