Identify outliers using mean and standard deviation

3 The marks \(x\) scored by a sample of 56 students in an examination are summarised by $$n = 56 , \quad \Sigma x = 3026 , \quad \Sigma x ^ { 2 } = 178890 .$$

1. Calculate the mean and standard deviation of the marks.
2. The highest mark scored by any of the 56 students in the examination was 93 . Show that this result may be considered to be an outlier.
3. The formula \(y = 1.2 x - 10\) is used to scale the marks. Find the mean and standard deviation of the scaled marks.

2 The marks \(x\) scored by a sample of 56 students in an examination are summarised by $$n = 56 , \quad \Sigma x = 3026 , \quad \Sigma x ^ { 2 } = 178890 .$$

1. Calculate the mean and standard deviation of the marks.
2. The highest mark scored by any of the 56 students in the examination was 93. Show that this result may be considered to be an outlier.
3. The formula \(y = 1.2 x - 10\) is used to scale the marks. Find the mean and standard deviation of the scaled marks.

1 Levels of nitrogen dioxide in the atmosphere are being monitored at the side of a road in a busy city centre. A sample of 18 measurements taken (in suitable units) is as follows. $$\begin{array} { l l l l l l l l l l l l l l l l l l } 83 & 44 & 95 & 92 & 98 & 63 & 69 & 76 & 19 & 91 & 70 & 91 & 74 & 65 & 62 & 70 & 95 & 108 \end{array}$$

1. Find the mean and standard deviation of the sample.
2. Hence identify, with justification, any possible outliers.

A sprinter runs many 100-metre trials, and the time, \(x\) seconds, for each is recorded. A sample of eight of these times is taken, as follows. 10.53 \quad 10.61 \quad 10.04 \quad 10.49 \quad 10.63 \quad 10.55 \quad 10.47 \quad 10.63

1. Calculate the sample mean, \(\bar{x}\), and sample standard deviation, \(s\), of these times. [3]
2. Show that the time of 10.04 seconds may be regarded as an outlier. [2]
3. Discuss briefly whether or not the time of 10.04 seconds should be discarded. [2]

The table below is an extract from the Large Data Set.

1. 1. Calculate the standard deviation of the engine sizes in the table. [1 mark]
   2. The mean of the engine sizes is 2084 Any value more than 2 standard deviations from the mean can be identified as an outlier. Using this definition of an outlier, show that the sample of engine sizes has exactly one outlier. Fully justify your answer. [3 marks]
2. Rajan calculates the mean of the masses of the cars in this extract and states that it is 1094 kg. Use your knowledge of the Large Data Set to suggest what error Rajan is likely to have made in his calculation. [1 mark]
3. Rajan claims there is an error in the data recorded in the table for one of the Toyotas from the South West, because there is no value for its carbon monoxide emissions. Use your knowledge of the Large Data Set to comment on Rajan's claim. [1 mark]

Amelia decides to analyse the heights of members of her school rowing club. The heights of a random sample of 10 rowers are shown in the table below.

Rower	Jess	Nell	Liv	Neve	Ann	Tori	Maya	Kath	Darcy	Jen
Height (cm)	162	169	172	156	146	161	159	164	157	160

1. Any value more than 2 standard deviations from the mean may be regarded as an outlier. Verify that Ann's height is an outlier. Fully justify your answer. [4 marks]
2. Amelia thinks she may have written down Ann's height incorrectly. If Ann's height were discarded, state with a reason what, if any, difference this would make to the mean and standard deviation. [2 marks]