Changing every value Problem set A Worked solutions

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

1. (a) The mean is 58 and the standard deviation is 9.5. Then 15 is added to every value.

   • Mean: 58 + 15 = 73
   • Standard deviation unchanged: 9.5

2. (b) The mean is 240 and the standard deviation is 31. Then 65 is subtracted from every value.

   • Mean: 240 − 65 = 175
   • Standard deviation unchanged: 31

Find the new median and IQR.

1. (a) A data set has median 63 and IQR 17. Every value is decreased by 25.

   • Median: 63 − 25 = 38
   • IQR unchanged: 17

2. (b) A data set has median 7.8 and IQR 2.4. Every value is increased by 3.5.

   • Median: 7.8 + 3.5 = 11.3
   • IQR unchanged: 2.4

Find the new median and IQR.

1. (a) A data set has median 14 and IQR 5. Every value is multiplied by 6.

   • Median: 14 × 6 = 84
   • IQR: 5 × 6 = 30

2. (b) A data set has median 90 and IQR 36. Every value is divided by 4.

   • Median: 90 ÷ 4 = 22.5
   • IQR: 36 ÷ 4 = 9

Find the new mean and standard deviation.

1. (a) The mean is 25 and the standard deviation is 4. Every value is multiplied by 3.

   • Mean: 25 × 3 = 75
   • SD: 4 × 3 = 12

2. (b) The mean is 7.5 and the standard deviation is 0.8. Every value is divided by 5.

   • Mean: 7.5 ÷ 5 = 1.5
   • SD: 0.8 ÷ 5 = 0.16

A data set has standard deviation 3.2. Every value is multiplied by 5. Find the new standard deviation and the new variance.

• New SD = 5 × 3.2 = 16
• New variance = 16² = 256
• Check: the old variance 3.2² = 10.24 is multiplied by 5² = 25: 256

The monthly pay of the workers in a factory, in euros, has mean 2800, standard deviation 350, median 2650 and IQR 400. Every worker gets a rise of 150 euros a month. Write down the new mean, standard deviation, median and IQR.

• Mean: 2800 + 150 = 2950
• Median: 2650 + 150 = 2800
• SD and IQR measure spread, so they do not change: 350 and 400

The daily temperatures in a town have mean 22 °C and standard deviation 3.5 °C. To change a temperature to °F, multiply by 1.8 and add 32. Find the mean and standard deviation in °F.

• Mean: 1.8 × 22 + 32 = 71.6
• SD: 1.8 × 3.5 = 6.3
• Adding 32 does not change the standard deviation

A teacher records how many minutes five students spend on homework: 14, 18, 19, 23, 26. Each student's time is doubled by the end of term and then 3 minutes are added.

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

   • From the calculator: x̄ = 20, σx = 4.15

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

   • New data: 31, 39, 41, 49, 55
   • From the calculator: x̄ = 43, σx = 8.29

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

   • Mean: 2 × 20 + 3 = 43
   • σ: 2 × 4.15 = 8.29
   • 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 12 and the original standard deviation is 3. The new mean is 39 and the new standard deviation is 9. Find k and b.

• New SD = k × 3 = 9, so k = 9 ÷ 3 = 3
• New mean = k × 12 + b = 39
• 3 × 12 + b = 39, so b = 39 − 36 = 3

The prices of the items in a shop have mean \$14 and standard deviation \$2. A sale takes 20% off every price. Delivery then adds \$3 to every item. Find the mean and standard deviation of the final prices.

• 20% off means multiplying by 0.8
• Mean: 0.8 × 14 + 3 = 14.2
• SD: 0.8 × 2 = 1.6 (adding 3 does not change it)