Calculate variance from summary statistics

1 The ages, \(x\) years, of 150 cars are summarised by \(\Sigma x = 645\) and \(\Sigma x ^ { 2 } = 8287.5\). Find \(\Sigma ( x - \bar { x } ) ^ { 2 }\), where \(\bar { x }\) denotes the mean of \(x\).

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.

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.

1 Janet and John wanted to compare their daily journey times to work, so they each kept a record of their journey times for a few weeks.

1. Janet's daily journey times, \(x\) minutes, for a period of 25 days, were summarised by \(\Sigma x = 2120\) and \(\Sigma x ^ { 2 } = 180044\). Calculate the mean and standard deviation of Janet's journey times.
2. John's journey times had a mean of 79.7 minutes and a standard deviation of 6.22 minutes. Describe briefly, in everyday terms, how Janet and John's journey times compare.

6 The maximum temperatures \(x\) degrees Celsius recorded during each month of 2005 in Cambridge are given in the table below. These data are summarised by \(n = 12 , \Sigma x = 180.6 , \Sigma x ^ { 2 } = 3107.56\).

1. Calculate the mean and standard deviation of the data.
2. Determine whether there are any outliers.
3. The formula \(y = 1.8 x + 32\) is used to convert degrees Celsius to degrees Fahrenheit. Find the mean and standard deviation of the 2005 maximum temperatures in degrees Fahrenheit.
4. In New York, the monthly maximum temperatures are recorded in degrees Fahrenheit. In 2005 the mean was 63.7 and the standard deviation was 16.0 . Briefly compare the maximum monthly temperatures in Cambridge and New York in 2005. The total numbers of hours of sunshine recorded in Cambridge during the month of January for each of the last 48 years are summarised below.

   Hours \(h\)	\(70 \leqslant h < 100\)	\(100 \leqslant h < 110\)	\(110 \leqslant h < 120\)	\(120 \leqslant h < 150\)	\(150 \leqslant h < 170\)	\(170 \leqslant h < 190\)
   Number of years	6	8	10	11	10	3

5. Draw a cumulative frequency graph for these data.
6. Use your graph to estimate the 90th percentile.

1 The maximum temperatures \(x\) degrees Celsius recorded during each month of 2005 in Cambridge are given in the table below. These data are summarised by \(n = 12 , \Sigma x = 180.6 , \Sigma x ^ { 2 } = 3107.56\).

1. Calculate the mean and standard deviation of the data.
2. Determine whether there are any outliers.
3. The formula \(y = 1.8 x + 32\) is used to convert degrees Celsius to degrees Fahrenheit. Find the mean and standard deviation of the 2005 maximum temperatures in degrees Fahrenheit.
4. In New York, the monthly maximum temperatures are recorded in degrees Fahrenheit. In 2005 the mean was 63.7 and the standard deviation was 16.0 . Briefly compare the maximum monthly temperatures in Cambridge and New York in 2005. The total numbers of hours of sunshine recorded in Cambridge during the month of January for each of the last 48 years are summarised below.

   Hours \(h\)	\(70 \leqslant h < 100\)	\(100 \leqslant h < 110\)	\(110 \leqslant h < 120\)	\(120 \leqslant h < 150\)	\(150 \leqslant h < 170\)	\(170 \leqslant h < 190\)
   Number of years	6	8	10	11	10	3

5. Draw a cumulative frequency graph for these data.
6. Use your graph to estimate the 90th percentile.

1. A disc of radius 1 cm is rolled onto a horizontal grid of rectangles so that the disc is equally likely to land anywhere on the grid. Each rectangle is 5 cm long and 3 cm wide. There are no gaps between the rectangles and the grid is sufficiently large so that no discs roll off the grid.

If the disc lands inside a rectangle without covering any part of the edges of the rectangle then a prize is won. By considering the possible positions for the centre of the disc,

1. show that the probability of winning a prize on any particular roll is \(\frac { 1 } { 5 }\) A group of 15 students each roll the disc onto the grid twenty times and record the number of times, \(x\), that each student wins a prize. Their results are summarised as follows $$\sum x = 61 \quad \sum x ^ { 2 } = 295$$
2. Find the standard deviation of the number of prizes won per student. A second group of 12 students each roll the disc onto the grid twenty times and the mean number of prizes won per student is 3.5 with a standard deviation of 2
3. Find the mean and standard deviation of the number of prizes won per student for the whole group of 27 students. The 27 students also recorded the number of times that the disc covered a corner of a rectangle and estimated the probability to be 0.2216 (to 4 decimal places).
4. Explain how this probability could be used to find an estimate for the value of \(\pi\) and state the value of your estimate.

1. Each member of a group of 27 people was timed when completing a puzzle.

The time taken, \(x\) minutes, for each member of the group was recorded.
These times are summarised in the following box and whisker plot. \includegraphics[max width=\textwidth, alt={}, center]{2b63aa7f-bc50-4422-8dc0-e661b521c221-08_353_1436_458_319}

1. Find the range of the times.
2. Find the interquartile range of the times. For these 27 people \(\sum x = 607.5\) and \(\sum x ^ { 2 } = 17623.25\)
3. calculate the mean time taken to complete the puzzle,
4. calculate the standard deviation of the times taken to complete the puzzle. Taruni defines an outlier as a value more than 3 standard deviations above the mean.
5. State how many outliers Taruni would say there are in these data, giving a reason for your answer. Adam and Beth also completed the puzzle in \(a\) minutes and \(b\) minutes respectively, where \(a > b\).
   When their times are included with the data of the other 27 people
   • the median time increases
   • the mean time does not change
   • Suggest a possible value for \(a\) and a possible value for \(b\), explaining how your values satisfy the above conditions.
   • Without carrying out any further calculations, explain why the standard deviation of all 29 times will be lower than your answer to part (d).

2 The continuous random variable \(Y\) has probability density function \(\mathrm { f } ( y )\) where $$\int _ { - \infty } ^ { \infty } y \mathrm { f } ( y ) \mathrm { d } y = 16 \text { and } \int _ { - \infty } ^ { \infty } y ^ { 2 } \mathrm { f } ( y ) \mathrm { d } y = 1040$$ Find the standard deviation of \(Y\) Circle your answer.
[0pt] [1 mark]
28
32
784
1024

A company sells sugar in bags which are labelled as containing 450 grams. Although the mean weight of sugar in a bag is more than 450 grams, there is concern that too many bags are underweight. The company can adjust the mean or the standard deviation of the weight of sugar in a bag.

1. State two adjustments the company could make. [2]

The weights, \(x\) grams, of a random sample of 25 bags are now recorded.

1. Given that \(\sum x = 11409\) and \(\sum x^2 = 5206937\), calculate the sample mean and sample standard deviation of these weights. [3]

Given that \(\sum x = 364\), \(\sum x^2 = 19412\), \(n = 10\), find \(\sigma\), the standard deviation of \(X\). Circle your answer. 24.8 \quad 44.1 \quad 616.2 \quad 1941.2 [1 mark]