Spread and standard deviation Problem set A Worked solutions

Find the mean absolute deviation of 7, 9, 12, 14, 18.

• Mean = 60 ÷ 5 = 12
• Distances from the mean: 5, 3, 0, 2, 6
• MAD = 16 ÷ 5 = 3.2

The deviations from the mean of a data set of five values are −6, −2, 1, 3, 4.

1. (a) Find the variance.

   • Squares: 36, 4, 1, 9, 16
   • Sum = 66
   • Variance = 66 ÷ 5 = 13.2

2. (b) Find the standard deviation.

   • σ = √13.2 = 3.63

The rainfall in five months, in mm, was 28, 35, 21, 40, 31.

1. (a) Find the mean.

   • Mean = 155 ÷ 5 = 31

2. (b) Complete a table of deviations and squared deviations, and find the sum of the squared deviations.

   • Deviations x − mean: −3, 4, −10, 9, 0
   • Squared deviations: 9, 16, 100, 81, 0
   • Sum = 206

3. (c) Find the standard deviation.

   • Variance = 206 ÷ 5 = 41.2
   • σ = √41.2 = 6.42

Two bus routes were timed. The dot plot shows the waiting times, in minutes, of eight passengers on each route. Route 1: 8, 9, 10, 10, 10, 10, 11, 12 Route 2: 4, 6, 8, 10, 10, 12, 14, 16

1. (a) Show that both have the same mean.

   • Route 1: sum = 80, 80 ÷ 8 = 10
   • Route 2: sum = 80, 80 ÷ 8 = 10

2. (b) Use your calculator to find the standard deviation of each.

   • From the calculator: Route 1 σx = 1.12, Route 2 σx = 3.74

3. (c) Which is more spread out? Give a reason.

   • Route 2 has the larger standard deviation
   • Its values lie further from the mean, as the dot plot shows

The scores in a game are roughly bell-shaped with mean 64 and standard deviation 5.

1. (a) What percentage lie between 59 and 69?

   • 59 = 64 − 5 and 69 = 64 + 5: 1 SD either side
   • About 68%

2. (b) What percentage lie between 54 and 74?

   • 54 and 74 are 2 SDs from the mean
   • About 95%

3. (c) What percentage lie above 79?

   • 79 is 3 SDs above the mean
   • About 99.7% lie within 3 SDs, so 0.3% lie outside
   • 0.3% ÷ 2 = 0.15% above

Five students did as many push-ups as they could in one minute: 40, 44, 41, 47, 43. The school champion did 54.

1. (a) Find the mean.

   • Mean = 215 ÷ 5 = 43

2. (b) Find the standard deviation.

   • Deviations: −3, 1, −2, 4, 0
   • Squares add to 30
   • σ = √(30 ÷ 5) = 2.45

3. (c) How many standard deviations above the mean is the champion?

   • Distance from the mean = 11
   • 11 ÷ 2.45 = 4.49

The heights of five seedlings, in cm, are 12, 15, 17, 20, 21.

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

   • From the calculator: x̄ = 17, σx = 3.29

2. (b) Each height is measured from the top of an 8 cm pot instead of from the soil, so 8 is added to every height. Find the new mean and standard deviation.

   • New values: 20, 23, 25, 28, 29
   • From the calculator: x̄ = 25, σx = 3.29

3. (c) After a month, every seedling is exactly 2 times as tall as it was at the start. Find the new mean and standard deviation.

   • New values: 24, 30, 34, 40, 42
   • From the calculator: x̄ = 34, σx = 6.57

4. (d) Describe what happens to the mean and to the standard deviation in each case.

   • Adding 8: mean + 8, σ unchanged (every value moves the same way, so the distances from the mean stay the same)
   • Multiplying by 2: mean × 2, σ × 2

The masses of eight parcels, in kg, are 4.2, 4.5, 4.6, 4.8, 5, 5.2, 5.3, 5.6. The last mass is typed into a computer as 56 instead of 5.6.

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

   • From the calculator: x̄ = 4.9, σx = 0.433

2. (b) Find the mean and standard deviation of the data with the error.

   • From the calculator: x̄ = 11.2, σx = 16.9

3. (c) Find the median and IQR of both sets of data.

   • Correct: Q1 = 4.55, Q3 = 5.25, IQR = 0.7
   • With error: Q1 = 4.55, Q3 = 5.25, IQR = 0.7

4. (d) Explain which measures of average and spread you would use for the data with the error.

   • The mean and SD use every value, so one extreme value drags them. The median and IQR depend on position, so they hardly notice it.

Two machines fill bottles. The volumes of five bottles from each machine, in ml, are Machine 1: 496, 498, 500, 502, 504 Machine 2: 494, 500, 500, 500, 506

1. (a) Show that both have the same mean and the same mean absolute deviation.

   • Both sums are 2500, so both means are 500
   • Distances: 4, 2, 0, 2, 4 and 6, 0, 0, 0, 6
   • Both sums of distances are 12: MAD = 2.4

2. (b) Find the standard deviation of each.

   • Machine 1: squares 16, 4, 0, 4, 16, σ = √(40 ÷ 5) = 2.83
   • Machine 2: squares 36, 0, 0, 0, 36, σ = √(72 ÷ 5) = 3.79

3. (c) Which is more consistent? Explain why MAD did not show the difference.

   • Machine 2 has two values far from the mean, and squaring makes those count more
   • Machine 1 has the smaller σ, so it is more consistent

The marks of 200 students are roughly bell-shaped with mean 70 and standard deviation 8.

1. (a) About how many have a value between 62 and 78?

   • 62 and 78 are 1 SD from the mean: 68%
   • 68% of 200 = 136

2. (b) About how many have a value above 86?

   • 86 is 2 SDs above the mean
   • 5% lie outside 2 SDs, so 2.5% lie above
   • 2.5% of 200 = 5

3. (c) Between which two values do the middle 95% lie?

   • 95% lie within 2 SDs of the mean
   • 70 − 2 × 8 = 54 and 70 + 2 × 8 = 86