AI SL · 13.3
0 of 44 skills done
Changing every value
13.3.3
Screenshot and video to come
The monthly pay of the staff at a firm has mean 3 400 and standard deviation 420 (in euros). Every person gets a rise of 250 a month. Write down the new mean and the new standard deviation.
Have a go first. Then open the working one line at a time.
New mean = 3 400 + 250 = 3 650
Every value goes up by 250, so the mean goes up by 250.
New standard deviation = 420
Every value and the mean move up together. Each distance from the mean stays the same, so σ is unchanged.
Mean 3 650, standard deviation 420
Adding k changes the centre and not the spread.
The data set 2, 4, 7, 9, 13 has mean 7 and standard deviation 3.85. Add 7 to every value. Find the new mean and standard deviation, and check the standard deviation with the deviations.
Have a go first. Then open the working one line at a time.
9, 11, 14, 16, 20
Add 7 to each value.
Deviations before and after are the same: −5, −3, 0, 2, 6
The mean moved from 7 to 14, the same 7 as every value.
New data: x̄ = 14, σx = 3.85
The mean is up by 7. σ is the same as before: 3.85.
The times, in seconds, that runners take to finish a hill climb have median 84 and IQR 22. Every runner's time is reduced by 30 seconds after a shorter route is used. Write down the new median and IQR.
Have a go first. Then open the working one line at a time.
84 − 30 = 54
The median is a value in the middle of the data, so it moves with the data.
IQR = 22
IQR = Q3 − Q1. Both quartiles go down by 30, so their difference does not change.
The data set 2, 4, 7, 9, 13 has mean 7, variance 14.8 and standard deviation 3.85. Every value is multiplied by 3. Find the new mean, variance and standard deviation.
Have a go first. Then open the working one line at a time.
6, 12, 21, 27, 39
Multiply each value by 3.
New mean = 3 × 7 = 21
The mean is multiplied by 3.
New σ = 3 × 3.847 = 11.5
Every distance from the mean is 3 times as long, so σ is multiplied by 3.
New variance = 3² × 14.8 = 133
Variance is a squared quantity, like the area of a square. Lengths × 3 means areas × 3² = 9. Check from the calculator: σx = 11.5.
The masses of some fruit, in grams, have median 360 and IQR 84. Every mass is divided by 4. Write down the new median and IQR.
Have a go first. Then open the working one line at a time.
360 ÷ 4 = 90
The median is divided by 4.
84 ÷ 4 = 21
Q1 and Q3 are both divided by 4, so their difference is divided by 4.
The temperatures in a town have mean 22 °C and standard deviation 3.5 °C. A temperature in °C is changed to °F by multiplying by 1.8 and then adding 32. Find the mean and standard deviation in °F.
Have a go first. Then open the working one line at a time.
New mean = 1.8 × 22 + 32 = 71.6
The mean follows the whole rule: multiply, then add.
New σ = 1.8 × 3.5 = 6.3
Multiplying by 1.8 multiplies σ by 1.8. Adding 32 does nothing to σ.
Mean 71.6 °F, standard deviation 6.3 °F
If the new data is ax + b, the new mean is a × mean + b and the new σ is a × σ.