Spread and standard deviation Problem set A

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

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

1. (a) Find the variance.
2. (b) Find the standard deviation.

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

1. (a) Find the mean.
2. (b) Complete a table of deviations and squared deviations, and find the sum of the squared deviations.
3. (c) Find the standard deviation.

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.
2. (b) Use your calculator to find the standard deviation of each.
3. (c) Which is more spread out? Give a reason.

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?
2. (b) What percentage lie between 54 and 74?
3. (c) What percentage lie above 79?

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.
2. (b) Find the standard deviation.
3. (c) How many standard deviations above the mean is the champion?

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

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.
2. (b) Find the mean and standard deviation of the data with the error.
3. (c) Find the median and IQR of both sets of data.
4. (d) Explain which measures of average and spread you would use for the data with the error.

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.
2. (b) Find the standard deviation of each.
3. (c) Which is more consistent? Explain why MAD did not show the difference.

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?
2. (b) About how many have a value above 86?
3. (c) Between which two values do the middle 95% lie?