Representation of data Banker

The times, in minutes, taken by 15 students in Karachi to travel to college one morning are shown in the stem-and-leaf diagram.1 | 2 5 82 | 1 2 4 5 7 83 | 1 3 6 84 | 25 | 5Key: 2 | 4 means 24 minutes

1. (a) Find the median and the interquartile range of the times. [3]
2. (b) Find the mean time, correct to 3 significant figures. [1]
3. (c) Give a reason why the median may be a better average than the mean for these data. [1]

The marks out of 50 scored in a quiz by 11 students in class P and 11 students in class Q are shown in the back-to-back stem-and-leaf diagram. Leaves for class P are on the left.8 | 1 |9 5 2 | 2 | 1 67 4 3 1 | 3 | 0 2 5 8 87 4 0 | 4 | 1 3 5 9Key: 5 | 2 | 6 means 25 marks for a student in class P and 26 marks for a student in class Q

1. (a) Find the median and the interquartile range for each class. [4]
2. (b) Make two comparisons between the marks of the two classes. [2]

The battery life, in minutes, of 19 phones of one model was measured during a video-streaming test.31 | 232 | 533 | 834 | 1 635 | 2 5 936 | 1 4 837 | 2 738 | 339 | 0 840 | 541 | 42 | 143 | 6Key: 35 | 9 means 359 minutes

1. (a) Find the range and the median of the battery lives. [2]
2. (b) Find the interquartile range. [2]
3. (c) Find the percentage of these phones with a battery life of more than 380 minutes, correct to 1 decimal place. [1]

A botanist in Murree measured the heights, in cm, of 19 young pine seedlings after one year. The results are shown below.3 | 44 | 1 5 7 85 | 2 3 5 6 8 96 | 1 2 4 6 77 | 1 48 | 9 | 6Key: 5 | 3 means 53 cm

1. (a) Find the median and the quartiles of the heights. [3]
2. (b) An outlier is defined as any value more than 1.5 × IQR above the upper quartile or below the lower quartile. Determine whether there are any outliers, showing your working. [3]
3. (c) Suggest one reason why the researcher might keep the value 96 in the data. [1]

1. (a) The stem-and-leaf diagram shows the masses, in grams, of 15 guavas from a market stall, where x is an unknown digit.3 | 2 5 84 | 0 1 3 x 6 7 95 | 0 2 4 7 8Key: 4 | 1 means 41 gGiven that the mean mass is 45.8 g, find the value of x. [3]
2. (b) Find the interquartile range of the masses. [2]

The rainfall, in mm, recorded on 19 days in August at a weather station in Abbottabad was:23 8 14 31 17 26 12 9 19 35 22 16 28 11 40 25 18 13 21

1. (a) Represent the data in a stem-and-leaf diagram with stems 0, 1, 2, 3, 4 and a key. [3]
2. (b) Find the median and the interquartile range. [3]

The resting pulse rates, in beats per minute, of 15 athletes and 15 non-athletes at a sports club in Faisalabad are shown. Leaves for athletes are on the left.8 | 4 |8 8 6 5 3 2 | 5 |9 6 4 3 1 0 | 6 | 2 5 84 1 | 7 | 0 1 2 4 5 6 8 9| 8 | 1 4 8| 9 | 3Key: 8 | 5 | 2 means 58 for an athlete and 52 for a non-athlete

1. (a) Find the median and the quartiles for each group. [4]
2. (b) Compare the resting pulse rates of the two groups in two different ways. [2]
3. (c) State one advantage of a back-to-back stem-and-leaf diagram over two box-and-whisker plots for these data. [1]

1. (a) The ages, in years, of 11 members of a cricket club committee are, in order,22 25 28 31 33 3a 41 4b 50 54 59where a and b are unknown digits. The median age is 37 years and the mean age is 39 years. Find a and b. [3]
2. (b) Find the interquartile range of the ages. [2]

The times, t seconds, taken by 100 people to solve a puzzle at a Lahore science fair are summarised below.Time t (s): 0 ≤ t < 20 | 20 ≤ t < 30 | 30 ≤ t < 40 | 40 ≤ t < 60 | 60 ≤ t < 100Frequency: 14 | 22 | 30 | 24 | 10

1. (a) Calculate the frequency density for each class, ready to draw a histogram. [3]
2. (b) Estimate the number of people who took more than 50 seconds. [2]

The lengths, l mm, of 141 leaves from a neem tree are summarised below.20 ≤ l < 25: 1225 ≤ l < 30: 2830 ≤ l < 35: 4535 ≤ l < 40: 1640 ≤ l < 50: 3250 ≤ l < 60: 8

1. (a) Calculate the frequency densities needed to draw a histogram of these data. [3]
2. (b) Estimate the number of leaves with lengths between 33 mm and 45 mm. [2]

A histogram shows the masses, m kg, of parcels handled by a courier office in Peshawar in one day. The classes and bar heights are:0 ≤ m < 2: height 3 cm2 ≤ m < 3: height 8 cm3 ≤ m < 5: height 5 cm5 ≤ m < 10: height 1.2 cmThe class 2 ≤ m < 3 contains 40 parcels.

1. (a) Find the frequency of each class and the total number of parcels. [4]
2. (b) Estimate the number of parcels with masses between 2.5 kg and 4 kg. [2]

The daily electricity use, x kWh, of 200 households in Lahore is summarised in the cumulative frequency table.x ≤ 5: 12x ≤ 10: 46x ≤ 15: 108x ≤ 20: 160x ≤ 25: 188x ≤ 30: 200(No household used 0 kWh or less.)

1. (a) Estimate the median daily use, assuming the points of the cumulative frequency graph are joined with straight lines. [2]
2. (b) Estimate the interquartile range. [3]
3. (c) Estimate the number of households that used more than 22 kWh. [2]

The ages, a years, of 80 visitors to a museum in Lahore one morning are shown in a histogram. The frequency densities (heights) of some bars and the frequencies of some classes are known:0 ≤ a < 10: frequency 810 ≤ a < 20: bar height 1.620 ≤ a < 40: frequency 24, bar height 1.240 ≤ a < 70: frequency 1870 ≤ a < 90: frequency unknown

1. (a) Find the frequency of the class 10 ≤ a < 20 and the frequency density of the class 70 ≤ a < 90. [4]
2. (b) State which class contains the median age. [1]

The typing speeds, w words per minute, of 120 applicants for an office job in Rawalpindi are summarised in a cumulative frequency table. The slowest applicant typed more than 10 words per minute. Assume the points of the cumulative frequency graph are joined by straight lines.w ≤ 20: 6w ≤ 30: 20w ≤ 40: 52w ≤ 50: 90w ≤ 60: 110w ≤ 70: 120

1. (a) Estimate the number of applicants with speeds of more than 45 words per minute. [2]
2. (b) The fastest 20% of applicants are invited to an interview. Estimate the least speed needed for an interview. [3]
3. (c) Estimate the 10th percentile of the speeds. [2]

100 students at a college in Multan recorded the number of text messages they sent on one day.Number of messages: 1–5 | 6–10 | 11–20 | 21–40Number of students: 15 | 25 | 40 | 20

1. (a) State the class boundaries of the class 11–20 and find the frequency densities for all four classes. [3]
2. (b) Estimate the number of students who sent more than 30 messages. [2]
3. (c) Calculate an estimate of the mean number of messages. [2]

1. (a) The lengths, x cm, of 20 fish caught at Keenjhar Lake are summarised by Σ(x − 50) = 84 and Σ(x − 50)² = 2400. Find the mean and the standard deviation of the lengths. [3]

A sample of 12 mangoes from orchard A in Multan has mean mass 250 g and standard deviation 8 g. A sample of 18 mangoes from orchard B has mean mass 238 g and standard deviation 6 g.

1. (a) Find Σx and Σx² for each orchard. [2]
2. (b) Find the mean and the standard deviation of the masses of all 30 mangoes. [4]

1. (a) For a set of 10 values of x, Σx = 360 and Σx² = 13 500. Find Σ(x − 30)². [2]
2. (b) Find the standard deviation of x, correct to 3 significant figures, and state the standard deviation of the values 5x + 2. [3]

1. (a) A set of n values of x has Σ(x − 10) = 48 and Σ(x − 10)² = 400. The mean of x is 12.4. Find n. [2]
2. (b) Find the standard deviation of x. [2]
3. (c) Find Σx². [2]

The times, x seconds, taken by 25 runners at a school sports day in Quetta to run 200 m were recorded. The mean was 40.8 s and the standard deviation was 5.2 s. It was later found that one time had been recorded as 52 s when it should have been 42 s.

1. (a) Find Σx and Σx² for the 25 recorded values. [2]
2. (b) Find the correct mean and the correct standard deviation. [4]

40 students in Hyderabad recorded the time, t hours, they spent on homework in one week.Time t: 0 ≤ t < 2 | 2 ≤ t < 4 | 4 ≤ t < 6 | 6 ≤ t < 10Frequency: 6 | 14 | 12 | 8

1. (a) Calculate estimates of the mean and the standard deviation of the time spent. [4]
2. (b) Explain why your answers are only estimates. [1]

The speeds, v km/h, of 100 cars on the M2 motorway were recorded by a speed camera.Speed v: 60 ≤ v < 80 | 80 ≤ v < 100 | 100 ≤ v < 110 | 110 ≤ v < 120 | 120 ≤ v < 140Frequency: 8 | 22 | 35 | 25 | 10

1. (a) Calculate estimates of the mean and the standard deviation of the speeds. [4]
2. (b) The speed limit on this part of the motorway is 120 km/h. Estimate the percentage of these cars exceeding the limit, and comment on whether the mean or the percentage is more useful to the police. [2]

The masses of 100 eggs from a poultry farm near Sheikhupura were measured to the nearest gram.Mass (g): 45–49 | 50–54 | 55–59 | 60–64 | 65–69Frequency: 9 | 23 | 38 | 21 | 9

1. (a) Write down the midpoints of the classes, explaining why the first midpoint is 47. [2]
2. (b) Calculate estimates of the mean and standard deviation of the masses. [3]

The distances, d km, travelled to work by the employees of a company in Islamabad are summarised below.Distance d: 10 ≤ d < 20 | 20 ≤ d < 30 | 30 ≤ d < 40 | 40 ≤ d < 60Frequency: 5 | k | 20 | 12

1. (a) Given that an estimate of the mean distance is 32.5 km, find k. [3]
2. (b) Calculate an estimate of the standard deviation of the distances. [3]

A call centre in Karachi recorded the durations, t minutes, of 100 calls.Duration t: 0 ≤ t < 2 | 2 ≤ t < 5 | 5 ≤ t < 10 | 10 ≤ t < 20 | 20 ≤ t < 30Frequency: 18 | 30 | 26 | 16 | 10

1. (a) Calculate estimates of the mean and the standard deviation of the call durations. [4]
2. (b) Estimate the number of calls that lasted longer than the mean plus one standard deviation. [2]
3. (c) State, with a reason, whether the median would be less than or greater than the mean. [1]

The daily maximum temperatures, in °C, in Jacobabad over 31 days in June have the following summary: lowest 39, lower quartile 43, median 44, upper quartile 47, highest 54.

1. (a) Find the interquartile range and determine whether the highest temperature is an outlier, where an outlier is a value more than 1.5 × IQR above the upper quartile. [3]
2. (b) Describe the shape of the distribution, giving a reason. [2]

The numbers of customers served in each of 15 half-hour periods at a bank counter in Gujranwala were:0 | 5 81 | 2 4 5 7 82 | 0 1 3 4 6 93 | 34 | 7Key: 2 | 3 means 23 customers

1. (a) Find the five values needed to draw a box-and-whisker plot of the data. [3]
2. (b) Show that 47 is an outlier using the 1.5 × IQR rule, and state the value at which the upper whisker would end if outliers are plotted separately. [3]

Box-and-whisker plots summarise the waiting times, in minutes, at two clinics in Rawalpindi on one day.Clinic A: lowest 4, lower quartile 9, median 13, upper quartile 20, highest 35Clinic B: lowest 6, lower quartile 10, median 12, upper quartile 14, highest 18

1. (a) Compare the waiting times at the two clinics in two ways. [2]
2. (b) Clinic A saw 120 patients that day. Estimate how many of them waited between 9 and 20 minutes. [1]
3. (c) A manager says: 'At clinic B no patient waited more than 18 minutes, but at clinic A at least a quarter of patients waited longer than any patient at clinic B.' Use the box plots to decide whether this statement is justified. [2]

Two varieties of wheat were grown on 40 one-hectare plots each in Punjab. The yields, in tonnes per hectare, are summarised:Variety X: mean 3.42, standard deviation 0.31Variety Y: mean 3.58, standard deviation 0.62

1. (a) Compare the yields of the two varieties in two ways. [2]
2. (b) A farmer wants a reliable harvest every year. Suggest which variety the farmer should choose, giving a reason. [1]
3. (c) Estimate the total yield from 25 hectares of variety Y. Also state the mean and the standard deviation of the yields of variety Y in kilograms per hectare. [2]

The monthly incomes, in thousands of rupees, of 9 workers at factory P in Sialkot are:32 35 35 38 40 41 44 46 180

1. (a) Find the mean and the median of the monthly incomes of the workers at factory P. [2]
2. (b) Explain which average better represents a typical worker at factory P. [1]
3. (c) At factory Q, the monthly incomes of 9 workers have median 42 thousand rupees and mean 43 thousand rupees. A worker claims that the workers at factory P earn more because 54.6 > 43. Comment on this claim. [2]