Identify outliers using IQR rule

2 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. $$\begin{array} { l l l l l l l l } 10.53 & 10.61 & 10.04 & 10.49 & 10.63 & 10.55 & 10.47 & 10.63 \end{array}$$

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

1 At a certain stage of a football league season, the numbers of goals scored by a sample of 20 teams in the league were as follows. \(\begin{array} { l l l l l l l l l l l l l l l l l l l l l } 22 & 23 & 23 & 23 & 26 & 28 & 28 & 30 & 31 & 33 & 33 & 34 & 35 & 35 & 36 & 36 & 37 & 46 & 49 & 49 \end{array}\)

1. Calculate the sample mean and sample variance, \(s ^ { 2 }\), of these data.
2. The three teams with the most goals appear to be well ahead of the other teams. Determine whether or not any of these three pieces of data may be considered outliers.

4 At a certain stage of a football league season, the numbers of goals scored by a sample of 20 teams in the league were as follows. \(\begin{array} { l l l l l l l l l l l l l l l l l l l l } 22 & 23 & 23 & 23 & 26 & 28 & 28 & 30 & 31 & 33 & 33 & 34 & 35 & 35 & 36 & 36 & 37 & 46 & 49 & 49 \end{array}\)

1. Calculate the sample mean and sample variance, \(s ^ { 2 }\), of these data.
2. The three teams with the most goals appear to be well ahead of the other teams. Determine whether or not any of these three pieces of data may be considered outliers.

1 The stem and leaf diagram illustrates the weights in grams of 20 house sparrows. Key: \(\quad 27 \quad \mid \quad 7 \quad\) represents 27.7 grams

1. Find the median and interquartile range of the data.
2. Determine whether there are any outliers.

4. A restaurant owner is concerned about the amount of time customers have to wait before being served. He collects data on the waiting times, to the nearest minute, of 20 customers. These data are listed below.

1. Find the median and inter-quartile range of the waiting times. An outlier is an observation that falls either \(1.5 \times\) (inter-quartile range) above the upper quartile or \(1.5 \times\) (inter-quartile range) below the lower quartile.
2. Draw a boxplot to represent these data, clearly indicating any outliers.
3. Find the mean of these data.
4. Comment on the skewness of these data. Justify your answer.

4. In a study of how students use their mobile telephones, the phone usage of a random sample of 11 students was examined for a particular week. The total length of calls, \(y\) minutes, for the 11 students were $$17,23,35,36,51,53,54,55,60,77,110$$

1. Find the median and quartiles for these data. A value that is greater than \(Q _ { 3 } + 1.5 \times \left( Q _ { 3 } - Q _ { 1 } \right)\) or smaller than \(Q _ { 1 } - 1.5 \times \left( Q _ { 3 } - Q _ { 1 } \right)\) is defined as an outlier.
2. Show that 110 is the only outlier.
3. Using the graph paper on page 15 draw a box plot for these data indicating clearly the position of the outlier. The value of 110 is omitted.
4. Show that \(S _ { y y }\) for the remaining 10 students is 2966.9 These 10 students were each asked how many text messages, \(x\), they sent in the same week. The values of \(S _ { x x }\) and \(S _ { x y }\) for these 10 students are \(S _ { x x } = 3463.6\) and \(S _ { x y } = - 18.3\).
5. Calculate the product moment correlation coefficient between the number of text messages sent and the total length of calls for these 10 students. A parent believes that a student who sends a large number of text messages will spend fewer minutes on calls.
6. Comment on this belief in the light of your calculation in part (e). \includegraphics[max width=\textwidth, alt={}, center]{d5d000c7-de42-461a-ba05-6c8b2c333780-09_611_1593_297_178}

6. A travel agent sells holidays from his shop. The price, in \(\pounds\), of 15 holidays sold on a particular day are shown below. For these data, find

1. the mean and the standard deviation,
2. the median and the inter-quartile range. An outlier is an observation that falls either more than \(1.5 \times\) (inter-quartile range) above the upper quartile or more than \(1.5 \times\) (inter-quartile range) below the lower quartile.
3. Determine if any of the prices are outliers. The travel agent also sells holidays from a website on the Internet. On the same day, he recorded the price, \(\pounds x\), of each of 20 holidays sold on the website. The cheapest holiday sold was \(\pounds 98\), the most expensive was \(\pounds 2400\) and the quartiles of these data were \(\pounds 305 , \pounds 1379\) and \(\pounds 1805\). There were no outliers.
4. On graph paper, and using the same scale, draw box plots for the holidays sold in the shop and the holidays sold on the website.
5. Compare and contrast sales from the shop and sales from the website. \section*{END}

3 The table shows fuel economy figures in miles per gallon (mpg) for some new cars.

1. Find the median and quartiles for the mpg of these fifteen cars.
2. Use the values in part (i) to identify any cars for which the mpg is an outlier.

Basel is a keen learner of languages. He finds a website on which a large number of language tutors offer their services. Basel records the cost, in dollars, of a one hour lesson from a random sample of tutors. He puts the data into a computer program which gives the following summary statistics. Cost per 1 hour lesson Min. :10.0 1st Qu. :16.0 Median :17.2 Mean :19.8 3rd Qu. :21.0 Max. :40.0

1. Showing all calculations, comment on any outliers for the cost of a one hour lesson with a language tutor. [4]
2. Describe the skewness of the data and explain what it means in this context. [2]

Dafydd is also a keen learner of languages. He takes his own random sample of the cost, in dollars, for a one hour lesson. He produces the following box plot. \includegraphics{figure_6}

1. 1. What will happen to the mean if the outlier is removed?
   2. What will happen to the median if the outlier is removed? [2]
2. Compare and contrast the distributions of the cost of one hour language lessons for Dafydd's sample and Basel's sample. [2]

The table shows fuel economy figures in miles per gallon (mpg) for some new cars.

Car	A	B	C	D	E	F	G	H	I	J	K	L	M	N	O
Mpg	57	40	34	33	11	17	30	27	31	20	35	24	26	23	32

1. Find the median and quartiles for the mpg of these 15 cars. [2]
2. Use the values in part (a) to identify any cars for which the mpg is an outlier. [3]

The table shows fuel economy figures in miles per gallon (mpg) for some new cars.

1. Find the median and quartiles for the mpg of these 15 cars. [2]
2. Use the values in part (a) to identify any cars for which the mpg is an outlier. [3]