Calculate mean from coded sums

1 For \(n\) values of the variable \(x\), it is given that $$\Sigma ( x - 50 ) = 144 \quad \text { and } \quad \Sigma x = 944 .$$ Find the value of \(n\).

2 A summary of 40 values of \(x\) gives the following information: $$\Sigma ( x - k ) = 520 , \quad \Sigma ( x - k ) ^ { 2 } = 9640$$ where \(k\) is a constant.

1. Given that the mean of these 40 values of \(x\) is 34 , find the value of \(k\).
2. Find the variance of these 40 values of \(x\).

2 The following table shows the results of a survey to find the average daily time, in minutes, that a group of schoolchildren spent in internet chat rooms. The mean time was calculated to be 27.5 minutes.

1. Form an equation involving \(f\) and hence show that the total number of children in the survey was 26 .
2. Find the standard deviation of these times.

2 The heights, \(x \mathrm {~cm}\), of a group of 82 children are summarised as follows. $$\Sigma ( x - 130 ) = - 287 , \quad \text { standard deviation of } x = 6.9 .$$

1. Find the mean height.
2. Find \(\Sigma ( x - 130 ) ^ { 2 }\).

2 The heights, \(x \mathrm {~cm}\), of a group of young children are summarised by $$\Sigma ( x - 100 ) = 72 , \quad \Sigma ( x - 100 ) ^ { 2 } = 499.2 .$$ The mean height is 104.8 cm .

1. Find the number of children in the group.
2. Find \(\Sigma ( x - 104.8 ) ^ { 2 }\).

4 The monthly rental prices, \(
) x$, for 9 apartments in a certain city are listed and are summarised as follows. $$\Sigma ( x - c ) = 1845 \quad \Sigma ( x - c ) ^ { 2 } = 477450$$ The mean monthly rental price is \(
) 2205$.

1. Find the value of the constant \(c\).
2. Find the variance of these values of \(x\).
3. Another apartment is added to the list. The mean monthly rental price is now \(
   ) 2120.50$. Find the rental price of this additional apartment.

1 Kadijat noted the weights, \(x\) grams, of 30 chocolate buns. Her results are summarised by $$\Sigma ( x - k ) = 315 , \quad \Sigma ( x - k ) ^ { 2 } = 4022$$ where \(k\) is a constant. The mean weight of the buns is 50.5 grams.

1. Find the value of \(k\).
2. Find the standard deviation of \(x\).

1 For 10 values of \(x\) the mean is 86.2 and \(\Sigma ( x - a ) = 362\). Find the value of

1. \(\Sigma x\),
2. the constant \(a\).

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 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 \(x\).

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.

1 For \(n\) values of the variable \(x\), it is given that \(\Sigma ( x - 100 ) = 216\) and \(\Sigma x = 2416\). Find the value of \(n\).

1 Andy counts the number of emails, \(x\), he receives each day and notes that, over a period of \(n\) days, \(\Sigma ( x - 10 ) = 27\) and the mean number of emails is 11.5 . Find the value of \(n\).

3 The masses, \(m\) grams, of 52 apples of a certain variety were found and summarised as follows. $$n = 52 \quad \Sigma ( m - 150 ) = - 182 \quad \Sigma ( m - 150 ) ^ { 2 } = 1768$$

1. Find the mean and variance of the masses of these 52 apples.
2. Use your answers from part (i) to find the exact value of \(\Sigma m ^ { 2 }\). The masses of the apples are illustrated in the box-and-whisker plot below.
   \includegraphics[max width=\textwidth, alt={}, center]{b5ce3230-7528-439c-9e85-ef159a49cba3-3_250_1310_662_383}
3. How many apples have masses in the interval \(130 \leqslant m < 140\) ?
4. An 'outlier' is a data item that lies more than 1.5 times the interquartile range above the upper quartile, or more than 1.5 times the interquartile range below the lower quartile. Explain whether any of the masses of these apples are outliers.

2 A sm mary \(\mathbf { 6 }\) th sp esl, \(x \mathrm { k }\) lm etres \(\boldsymbol { \rho } \mathbf { r b }\), \(\mathbf { 0 } 2\) cars \(\boldsymbol { \rho }\) ssig a certain \(\dot { \mathrm { p } } \mathrm { ng }\) th fb low ig if o matin $$\Sigma ( x \oplus ) = 3 \mathrm { a } \quad \mathrm {~d} \quad \Sigma ( x \oplus ) ^ { 2 } = \mathbb { t }$$ Fid b riance 6 th sp ed ad \(n\) e fid b vale \(6 \Sigma x ^ { 2 }\). [4]

1. Stav is studying the large data set for September 2015

He codes the variable Daily Mean Pressure, \(x\), using the formula \(y = x - 1010\)
The data for all 30 days from Hurn are summarised by $$\sum y = 214 \quad \sum y ^ { 2 } = 5912$$

1. State the units of the variable \(x\)
2. Find the mean Daily Mean Pressure for these 30 days.
3. Find the standard deviation of Daily Mean Pressure for these 30 days. Stav knows that, in the UK, winds circulate
   • in a clockwise direction around a region of high pressure
   • in an anticlockwise direction around a region of low pressure
   The table gives the Daily Mean Pressure for 3 locations from the large data set on 26/09/2015

   Location	Heathrow	Hurn	Leuchars
   Daily Mean Pressure	1029	1028	1028
   Cardinal Wind Direction

   The Cardinal Wind Directions for these 3 locations on 26/09/2015 were, in random order, $$\begin{array} { l l l } W & N E & E \end{array}$$ You may assume that these 3 locations were under a single region of pressure.
4. Using your knowledge of the large data set, place each of these Cardinal Wind Directions in the correct location in the table.
   Give a reason for your answer. \section*{Question 3 continued.}

1. Jim records the length, \(l \mathrm {~mm}\), of 81 salmon. The data are coded using \(x = l - 600\) and the following summary statistics are obtained.

$$n = 81 \quad \sum x = 3711 \quad \sum x ^ { 2 } = 475181$$

1. Find the mean length of these salmon.
2. Find the variance of the lengths of these salmon. The weight, \(w\) grams, of each of the 81 salmon is recorded to the nearest gram. The recorded results for the 81 salmon are summarised in the box plot below.
   \includegraphics[max width=\textwidth, alt={}, center]{b8ac20db-4237-4def-81aa-a3eecbeefbdd-10_362_1479_849_296}
3. Find the maximum number of salmon that have weights in the interval $$4600 < w \leqslant 7700$$ Raj says that the box plot is incorrect as Jim has not included outliers.
   For these data an outlier is defined as a value that is more than
   \(1.5 \times\) IQR above the upper quartile or \(1.5 \times\) IQR below the lower quartile
4. Show that there are no outliers.

2 Before leaving for a tour of the UK during the summer of 2008, Eduardo was told that the UK price of a 1.5-litre bottle of spring water was about 50p. Whilst on his tour, Eduardo noted the prices, \(x\) pence, which he paid for 1.5-litre bottles of spring water from 12 retail outlets. He then subtracted 50 p from each price and his resulting differences, in pence, were $$\begin{array} { l l l l l l l l l l l l } - 18 & - 11 & 1 & 15 & 7 & - 1 & 17 & - 16 & 18 & - 3 & 0 & 9 \end{array}$$

1. 1. Calculate the mean and the standard deviation of these differences.
   2. Hence calculate the mean and the standard deviation of the prices, \(x\) pence, paid by Eduardo.
2. Based on an exchange rate of \(€ 1.22\) to \(\pounds 1\), calculate, in euros, the mean and the standard deviation of the prices paid by Eduardo.
   \includegraphics[max width=\textwidth, alt={}]{c4844a30-6a86-49e3-b6aa-8e213dfc8ca1-05_2484_1709_223_153}

1. Employees at a company were asked how long their average commute to work was. The table below gives information about their answers.

The company estimates that the mean time for employees commuting to work is 28 minutes. Work out the value of \(x\), showing your working clearly.
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2 Lizzie, the receptionist at a dental practice, was asked to keep a weekly record of the number of patients who failed to turn up for an appointment. Her records for the first 15 weeks were as follows. $$\begin{array} { l l l l l l l l l l l l l l l } 20 & 26 & 32 & a & 37 & 14 & 27 & 34 & 15 & 18 & b & 25 & 37 & 29 & 25 \end{array}$$ Unfortunately, Lizzie forgot to record the actual values for two of the 15 weeks, so she recorded them as \(a\) and \(b\). However, she did remember that \(a < 10\) and that \(b > 40\).

1. Calculate the median and the interquartile range of these 15 values.
2. Give a reason why, for these data:
   1. the mode is not an appropriate measure of average;
   2. the standard deviation cannot be used as a measure of spread.
3. Subsequent investigations revealed that the missing values were 8 and 43 . Calculate the mean and the standard deviation of the 15 values.