Spread and standard deviation Problem set B Worked solutions

Find the mean absolute deviation of 5, 8, 9, 15, 18.

• Mean = 55 ÷ 5 = 11
• Distances from the mean: 6, 3, 2, 4, 7
• MAD = 22 ÷ 5 = 4.4

The deviations from the mean of a data set of five values are −7, −1, 2, 3, 3.

1. (a) Find the variance.

   • Squares: 49, 1, 4, 9, 9
   • Sum = 72
   • Variance = 72 ÷ 5 = 14.4

2. (b) Find the standard deviation.

   • σ = √14.4 = 3.79

A basketball player scored 54, 47, 62, 41, 56 points in five games.

1. (a) Find the mean.

   • Mean = 260 ÷ 5 = 52

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

   • Deviations x − mean: 2, −5, 10, −11, 4
   • Squared deviations: 4, 25, 100, 121, 16
   • Sum = 266

3. (c) Find the standard deviation.

   • Variance = 266 ÷ 5 = 53.2
   • σ = √53.2 = 7.29

Two bakeries weighed eight loaves each. The dot plot shows the masses, in grams. Bakery 1: 497, 498, 500, 500, 500, 500, 502, 503 Bakery 2: 491, 494, 497, 500, 500, 503, 506, 509

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

   • Bakery 1: sum = 4000, 4000 ÷ 8 = 500
   • Bakery 2: sum = 4000, 4000 ÷ 8 = 500

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

   • From the calculator: Bakery 1 σx = 1.80, Bakery 2 σx = 5.61

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

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

The heights of a type of sunflower, in cm, are roughly bell-shaped with mean 140 and standard deviation 12.

1. (a) What percentage lie between 128 and 152?

   • 128 = 140 − 12 and 152 = 140 + 12: 1 SD either side
   • About 68%

2. (b) What percentage lie between 116 and 164?

   • 116 and 164 are 2 SDs from the mean
   • About 95%

3. (c) What percentage lie above 176?

   • 176 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 people each assembled a flat-pack kit. Their times, in minutes, were 62, 58, 65, 60, 60. The fastest known time is 54 minutes.

1. (a) Find the mean.

   • Mean = 305 ÷ 5 = 61

2. (b) Find the standard deviation.

   • Deviations: 1, −3, 4, −1, −1
   • Squares add to 28
   • σ = √(28 ÷ 5) = 2.37

3. (c) How many standard deviations below the mean is the fastest known time?

   • Distance from the mean = 7
   • 7 ÷ 2.37 = 2.96

Five journey times, in minutes, are 18, 22, 25, 27, 33.

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

   • From the calculator: x̄ = 25, σx = 5.02

2. (b) Roadworks add 6 minutes to every journey. Find the new mean and standard deviation.

   • New values: 24, 28, 31, 33, 39
   • From the calculator: x̄ = 31, σx = 5.02

3. (c) On a slow bus, every journey takes exactly 2 times as long as the original. Find the new mean and standard deviation.

   • New values: 36, 44, 50, 54, 66
   • From the calculator: x̄ = 50, σx = 10.0

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

   • Adding 6: mean + 6, σ unchanged (every value moves the same way, so the distances from the mean stay the same)
   • Multiplying by 2: mean × 2, σ × 2

Eight students swam one length of a pool. Their times, in seconds, are 31, 33, 34, 36, 37, 39, 40, 42. The last time is typed into a computer as 420 instead of 42.

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

   • From the calculator: x̄ = 36.5, σx = 3.5

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

   • From the calculator: x̄ = 83.8, σx = 127

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

   • Correct: Q1 = 33.5, Q3 = 39.5, IQR = 6
   • With error: Q1 = 33.5, Q3 = 39.5, IQR = 6

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 bakers cut baguettes. The lengths of five baguettes from each baker, in cm, are Baker 1: 37, 39, 40, 41, 43 Baker 2: 36, 40, 40, 40, 44

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

   • Both sums are 200, so both means are 40
   • Distances: 3, 1, 0, 1, 3 and 4, 0, 0, 0, 4
   • Both sums of distances are 8: MAD = 1.6

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

   • Baker 1: squares 9, 1, 0, 1, 9, σ = √(20 ÷ 5) = 2
   • Baker 2: squares 16, 0, 0, 0, 16, σ = √(32 ÷ 5) = 2.53

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

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

The masses of 400 lemons, in grams, are roughly bell-shaped with mean 120 and standard deviation 9.

1. (a) About how many have a value between 111 and 129?

   • 111 and 129 are 1 SD from the mean: 68%
   • 68% of 400 = 272

2. (b) About how many have a value above 138?

   • 138 is 2 SDs above the mean
   • 5% lie outside 2 SDs, so 2.5% lie above
   • 2.5% of 400 = 10

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

   • 95% lie within 2 SDs of the mean
   • 120 − 2 × 9 = 102 and 120 + 2 × 9 = 138