2.02g Calculate mean and standard deviation

4 The times taken, \(t\) seconds, by 1140 people to solve a puzzle are summarised in the table.

1. On the grid, draw a histogram to illustrate this information. \includegraphics[max width=\textwidth, alt={}, center]{7652f36c-59b5-4fcd-b17b-d796dc82aec0-05_812_1406_804_411}
2. Calculate an estimate of the mean of \(t\).

1 Rani and Diksha go shopping for clothes.

1. Rani buys 4 identical vests, 3 identical sweaters and 1 coat. Each vest costs \(\\) 5.50\( and the coat costs \)\\( 90\). The mean cost of Rani's 8 items is \(\\) 29\(. Find the cost of a sweater.
2. Diksha buys 1 hat and 4 identical shirts. The mean cost of Diksha's 5 items is \)\\( 26\) and the standard deviation is \(\\) 0\(. Explain how you can tell that Diksha spends \)\\( 104\) on shirts.

1 In a statistics lesson 12 people were asked to think of a number, \(x\), between 1 and 20 inclusive. From the results Tom found that \(\Sigma x = 186\) and that the standard deviation of \(x\) is 4.5. Assuming that Tom's calculations are correct, find the values of \(\Sigma ( x - 10 )\) and \(\Sigma ( x - 10 ) ^ { 2 }\).

5 The lengths, \(t\) minutes, of 242 phone calls made by a family over a period of 1 week are summarised in the frequency table below.

1. Find the value of \(a\).
2. Calculate an estimate of the mean length of these phone calls.
3. On the grid, draw a histogram to illustrate the data in the table. \includegraphics[max width=\textwidth, alt={}, center]{a813e127-d116-411c-88ec-2443fdbc9391-07_2002_1513_486_356}

4 Farfield Travel and Lacket Travel are two travel companies which arrange tours abroad. The numbers of holidays arranged in a certain week are recorded in the table below, together with the means and standard deviations of the prices.

1. Calculate the mean price of all 51 holidays.
2. The prices of individual holidays with Farfield Travel are denoted by \(\\) x _ { F }\( and the prices of individual holidays with Lacket Travel are denoted by \)\\( x _ { L }\). By first finding \(\Sigma x _ { F } ^ { 2 }\) and \(\Sigma x _ { L } ^ { 2 }\), find the standard deviation of the prices of all 51 holidays.

1 The times, \(t\) seconds, taken to swim 100 m were recorded for a group of 9 swimmers and were found to be as follows. $$\begin{array} { l l l l l l l l l } 95 & 126 & 117 & 135 & 120 & 125 & 114 & 119 & 136 \end{array}$$

1. Find the values of \(\Sigma ( t - 120 )\) and \(\Sigma ( t - 120 ) ^ { 2 }\).
2. Using your values found in part (i), calculate the variance of \(t\).

7 The times in minutes taken by 13 pupils at each of two schools in a cross-country race are recorded in the table below.

1. Draw a back-to-back stem-and-leaf diagram to illustrate these times with Thaters School on the left.
2. Find the interquartile range of the times for pupils at Thaters School.
   The times taken by pupils at Whitefay Park School are denoted by \(x\) minutes.
3. Find the value of \(\Sigma ( x - 60 ) ^ { 2 }\).
4. It is given that \(\Sigma ( x - 60 ) = 46\). Use this result, together with your answer to part (iii), to find the variance of \(x\).
   If you use the following lined page to complete the answer(s) to any question(s), the question number(s) must be clearly shown.

1 Twelve values of \(x\) are shown below. Find the mean and standard deviation of \(( x - 1760 )\). Hence find the mean and standard deviation of \(x\). [4]

2 For 40 values of the variable \(x\), it is given that \(\Sigma ( x - c ) ^ { 2 } = 3099.2\), where \(c\) is a constant. The standard deviation of these values of \(x\) is 3.2 .

1. Find the value of \(\Sigma ( x - c )\).
2. Given that \(c = 50\), find the mean of these values of \(x\).

1 A computer can generate random numbers which are either 0 or 2 . On a particular occasion, it generates a set of numbers which consists of 23 zeros and 17 twos. Find the mean and variance of this set of 40 numbers.

4 The ages, \(x\) years, of 18 people attending an evening class are summarised by the following totals: \(\Sigma x = 745 , \Sigma x ^ { 2 } = 33951\).

1. Calculate the mean and standard deviation of the ages of this group of people.
2. One person leaves the group and the mean age of the remaining 17 people is exactly 41 years. Find the age of the person who left and the standard deviation of the ages of the remaining 17 people.

4 A group of 10 married couples and 3 single men found that the mean age \(\bar { x } _ { w }\) of the 10 women was 41.2 years and the standard deviation of the women's ages was 15.1 years. For the 13 men, the mean age \(\bar { x } _ { m }\) was 46.3 years and the standard deviation was 12.7 years.

1. Find the mean age of the whole group of 23 people.
2. The individual women's ages are denoted by \(x _ { w }\) and the individual men's ages by \(x _ { m }\). By first finding \(\Sigma x _ { w } ^ { 2 }\) and \(\Sigma x _ { m } ^ { 2 }\), find the standard deviation for the whole group.

1 A summary of 24 observations of \(x\) gave the following information: $$\Sigma ( x - a ) = - 73.2 \quad \text { and } \quad \Sigma ( x - a ) ^ { 2 } = 2115 .$$ The mean of these values of \(x\) is 8.95 .

1. Find the value of the constant \(a\).
2. Find the standard deviation of these values of \(x\).

1 Rachel measured the lengths in millimetres of some of the leaves on a tree. Her results are recorded below. $$\begin{array} { l l l l l l l l l l } 32 & 35 & 45 & 37 & 38 & 44 & 33 & 39 & 36 & 45 \end{array}$$ Find the mean and standard deviation of the lengths of these leaves.

6 The following table gives the marks, out of 75, in a pure mathematics examination taken by 234 students.

1. Draw a histogram on graph paper to represent these results.
2. Calculate estimates of the mean mark and the standard deviation.

1 Anita made observations of the maximum temperature, \(t ^ { \circ } \mathrm { C }\), on 50 days. Her results are summarised by \(\Sigma t = 910\) and \(\Sigma ( t - \bar { t } ) ^ { 2 } = 876\), where \(\bar { t }\) denotes the mean of the 50 observations. Calculate \(\bar { t }\) and the standard deviation of the observations.

2 Esme noted the test marks, \(x\), of 16 people in a class. She found that \(\Sigma x = 824\) and that the standard deviation of \(x\) was 6.5.

1. Calculate \(\Sigma ( x - 50 )\) and \(\Sigma ( x - 50 ) ^ { 2 }\).
2. One person did the test later and her mark was 72. Calculate the new mean and standard deviation of the marks of all 17 people.

4 In a survey, the percentage of meat in a certain type of take-away meal was found. The results, to the nearest integer, for 193 take-away meals are summarised in the table.

1. Calculate estimates of the mean and standard deviation of the percentage of meat in these take-away meals.
2. Draw, on graph paper, a histogram to illustrate the information in the table.

3 Swati measured the lengths, \(x \mathrm {~cm}\), of 18 stick insects and found that \(\Sigma x ^ { 2 } = 967\). Given that the mean length is \(\frac { 58 } { 9 } \mathrm {~cm}\), find the values of \(\Sigma ( x - 5 )\) and \(\Sigma ( x - 5 ) ^ { 2 }\).

4 The following histogram summarises the times, in minutes, taken by 190 people to complete a race. \includegraphics[max width=\textwidth, alt={}, center]{df246a50-157b-49f7-bba0-f9b86960b8b9-2_1210_1125_1251_513}

1. Show that 75 people took between 200 and 250 minutes to complete the race.
2. Calculate estimates of the mean and standard deviation of the times of the 190 people.
3. Explain why your answers to part (ii) are estimates.

4 Barry weighs 20 oranges and 25 lemons. For the oranges, the mean weight is 220 g and the standard deviation is 32 g . For the lemons, the mean weight is 118 g and the standard deviation is 12 g .

1. Find the mean weight of the 45 fruits.
2. The individual weights of the oranges in grams are denoted by \(x _ { o }\), and the individual weights of the lemons in grams are denoted by \(x _ { l }\). By first finding \(\Sigma x _ { o } ^ { 2 }\) and \(\Sigma x _ { l } ^ { 2 }\), find the variance of the weights of the 45 fruits.

1 Find the mean and variance of the following data. $$\begin{array} { l l l l l l l l l l } 5 & - 2 & 12 & 7 & - 3 & 2 & - 6 & 4 & 0 & 8 \end{array}$$

2 A traffic camera measured the speeds, \(x\) kilometres per hour, of 8 cars travelling along a certain street, with the following results. $$\begin{array} { l l l l l l l l } 62.7 & 59.6 & 64.2 & 61.5 & 68.3 & 66.9 & 62.0 & 62.3 \end{array}$$

1. Find \(\Sigma ( x - 62 )\).
2. Find \(\Sigma ( x - 62 ) ^ { 2 }\).
3. Find the mean and variance of the speeds of the 8 cars.

4

1. Amy measured her pulse rate while resting, \(x\) beats per minute, at the same time each day on 30 days. The results are summarised below. $$\Sigma ( x - 80 ) = - 147 \quad \Sigma ( x - 80 ) ^ { 2 } = 952$$ Find the mean and standard deviation of Amy's pulse rate.
2. Amy's friend Marok measured her pulse rate every day after running for half an hour. Marok's pulse rate, in beats per minute, was found to have a mean of 148.6 and a standard deviation of 18.5. Assuming that pulse rates have a normal distribution, find what proportion of Marok's pulse rates, after running for half an hour, were above 160 beats per minute.

1 The time taken, \(t\) hours, to deliver letters on a particular route each day is measured on 250 working days. The mean time taken is 2.8 hours. Given that \(\Sigma ( t - 2.5 ) ^ { 2 } = 96.1\), find the standard deviation of the times taken.