2.1 Statistical Measures

The speeds (s), to the nearest mile per hour, of 80 vehicles passing a speed camera were recorded and are grouped in the table below. 

A veterinary nurse records the weight of puppies (in kg) at birth and again at their eight week check-up.  The table below summarises the weight gain of 50 small breed puppies over their first eight weeks.

Use linear interpolation to estimate the median and interquartile range of the weight gain of the 50 puppies.

Give a reason why it is not possible to determine the exact median for this data.

The veterinary nurse decides to monitor any puppies whose weight gain during their first 8 weeks was less than 0.8 kg. 

Estimate the number of puppies whose weight gain could be below 0.8 kg, and explain why the vet would need more information before determining for certain how many puppies would need to be monitored.

Workers at an elephant sanctuary measure the health of their elephants by weighing the amount of dung () each one produces.  The data for the mass of dung produced in one day by 18 elephants can be summarised as  = 895  and  ²= 45 810.

Calculate the mean and variance of the amount of dung produced that day.

To calculate how much food (f ) to give each elephant, staff use the formula f= 3 – 25.

Calculate the mean and variance of the amount of food the workers should give to the elephants each day.

A machine is set to fill sacks of potatoes to a target weight of 50 kg, although the actual weight of the sacks () can vary from that target.  

To test the accuracy of the machine, a random sample of 20 sacks is taken and the values of =(-50) are recorded.  

The mean and standard deviation of are found to be -1.8 and 3.1 respectively. 

Write down the mean and standard deviation of .

Calculate the value of

²

Another 10 sacks of potatoes are sampled and the mean of these is found to be 51.2 kg.  Calculate the mean of all 30 sacks of potatoes and comment on the accuracy of the machine.

Whilst in lockdown, 100 people were asked to record the length of time, rounded to the nearest minute, that they spent exercising on a particular day. 

The results are summarised in the table below:

Given that the estimate of the mean time spent exercising based on this table is 35.4 minutes, find the values of and .

The ages, years, of 200 people attending a vaccination clinic in one day are summarised by the following:  = 7211  and  = 275 360.

Calculate the mean and standard deviation of the ages of the people attending the clinic that day.

One person chooses not to get the vaccine, so their data is discounted. The new mean is exactly 36.  Calculate the age of the person who left and the standard deviation of the remaining people.

In January of 2021, the UK government announced a nationwide lockdown to control the spread of the coronavirus.  The table below shows the means and standard deviations of the average amounts of time spent indoors per day by some people in London, UK and in Wellington, New Zealand, in January of 2021.

Suggest a reason, in the context of the question, for why

Based on the data in the table, do you think the government in New Zealand had imposed the same restrictions as those in the UK?  Give a reason for your answer.

Calculate the overall mean for the average amounts of time spent indoors by all 40 people.

Data on the daily maximum gust in Camborne is gathered from the large data set.  For the third week of October 1987, and the same week in 2015, the daily maximum gust in knots (kn) is shown in the table below:

Using the data given above, find the mean and standard deviation of the daily maximum gust for Camborne for the third week of October 1987.

Without carrying out any calculations, state whether you think the mean and standard deviation for the 2015 data would be greater or less than your answers for part (a).  Use your knowledge of the large data set to suggest a reason for each of your answers.

Fran sits three Maths papers and six Science papers during her final A Level exams.  She achieves a mean score of 62.7% across the three Maths exam papers, and needs an overall mean score of 78.5% across all nine papers to get into her chosen University.  After getting the results of four out of her six Science papers, her mean score in Science is 84.2%.

Given that each of the nine papers is weighted equally when working out the mean scores, calculate the mean score she must achieve on her final two science papers in order to gain a place at University.

Coffee4Life manufactures reusable coffee cups out of coffee plant waste.  Coffee cups are tested to see how many times they can be used before they begin to disintegrate.  A sample of 15 cups are tested, giving the following results for numbers of uses:

                     31    36    41    43    47

                     49    51    56    58    62

                     62    63    68    69    72

The advertising department at Coffee4Life designs an advert which says;

“If used once a day,   of our cups last longer than 9 weeks.”

Explain the mistake that the advertising department has made, and state how the advert could be reworded to make it correct.

The lengths ( cm) of a sample of nine otters, measured to the nearest centimetre by a wildlife research team, are:

                        76     77      91      65       63      83      92      61      88

Calculate the mean and standard deviation of the nine recorded lengths.

A machine is set to fill sacks of potatoes to a weight of 50 kg.  In a random sample the following weights in kg were recorded:

                       53.4     46.8     47.2     42.4     43.0     47.4      55.2

Code the weights using the coding  =0.5 (weight in kg-40)

During initial training for the Royal Air Force new recruits must sit an aptitude test.  Test scores for the latest round of recruits are shown in the table below:

Recruits who score below the 25th percentile are disqualified.

Calculate an estimate for the score recruits must have achieved to avoid disqualification. 

Those who score in the top 30% move on to the next stage of training and the rest must re-sit the test.

One of the recruits, Amelia, achieves a score of 231.  Estimate whether Amelia will need to re-sit the test or will be moved on to the next stage of training. 

A random sample of 50 students were asked how long they spent revising for their Maths exam in the 24 hours before the exam.  The results are shown in the table below:

For this data, use linear interpolation to estimate the median.

Using  to represent the mid-point of each class, = 8340 and = 1 636 200.

Estimate the mean and the standard deviation of the amount of time students spent revising.

A college needs to standardise the test scores of all students before submitting them to the Exam Board.  The scores are standardised by using the coding , where  represents the raw test score and represents the standardised score.  The college calculates the mean standardised test score to be 0.74.  Find the mean of the raw test scores.

A selection of data from the large data set relating to the mean daily air temperature in Beijing for the first 10 days in October in both 1987 and 2015 is given above.  Climate activists use temperature data to track changes over time.

Using the data given above, find the mean of the daily mean air temperature for both 1987 and 2015.

Climate activists claim that temperature changes of more than 0.5°C each decade would indicate significant evidence of irreparable climate change damage.

Zisien measures the speeds,  miles per hour, of a number of cars passing her house one day.  She knows that the speed limit is 30 miles per hour so she decides to use the coding  =  – 30  when she records the data. 

She finds that  =13.4  and  = 1470.

Is it more likely that more of the cars in Zisien’s sample were going over or under the speed limit that day? Give a reason for your answer.

Given that the mean speed of the cars in Zisien’s sample, , is 30.67 miles per hour, calculate the standard deviation of .

Zisien’s sister, Ying, used the code =  – 20 to record the data for the same cars.

Calculate the value of  .

Two friends, Anna and Connor, are playing a gaming app on their phones.  As they play, they can choose from three different booster options.  They are unaware that each of the three options are charging them automatically from their mobile accounts.  The number of in-app purchases they each make are shown in the table below.

Botanists at a research centre are carrying out research on a new type of fertiliser.  They collect data on the heights of one group of geraniums growing without the fertiliser (the control group) and of another group growing with the fertiliser (the experimental group).  They take care to keep all other growing conditions the same for both groups.

The table below shows the heights of the control group of geraniums 15 weeks after planting. 

The data for the group of geraniums growing in the experimental group were summarised as follows:

 Q₁=23.4 cm                 Q₂=27.1 cm                 Q₃=28.5 cm

The shortest plant in the experimental group was 15.2 cm and the tallest was 33.5 cm. 

Compare the distribution of the heights of the plants in the two groups.

Wildlife researchers are studying the swimming speeds, kmph, of two species of penguin, the emperor penguin and the gentoo penguin.  The mean swimming speed of 40 gentoo penguins was found to be 31.4 kmph and the standard deviation was found to be 3.8 kmph.

Allowing to represent the swimming speeds of the gentoo penguins, show that =1256  and calculate the value of  .

The swimming speeds of 20 emperor penguins () were also recorded and the mean swimming speed of all 60 penguins surveyed was found to be 24.1 kmph. Given that  =41891,  calculate the mean and standard deviation of the 20 emperor penguins.

Some entomologists were studying the amount of time two different species of butterflies spent cocooned.  The table shows the means and standard deviations of the time spent cocooned, measured in days, by 15 Monarch butterflies and 25 Common Blue butterflies.

Given that the overall mean time for all 40 butterflies was 11.93 days, calculate the mean number of days the Monarch butterflies spent cocooned and complete the table.

Calculate the overall standard deviation of the time spent cocooned by all 40 butterflies.

Lab technicians were studying the effect of caffeine on mice.  The resting heart rates,  beats per minute (bpm), of some mice were recorded and the results were summarised by  = 150  and   = 1050,  where is a constant.

Given that the variance of the resting heart rates was found to be 10 bpm², calculate the two possible options for the number of mice in the study.

The mean resting heart rate is found to be 605 bpm.  Using this information, write down the two possible options for the value of a.

Hattie’s homeroom teacher decides to summarise the number of minutes, t, she has been late to school during the last year in preparation for a parents’ meeting.  The results are shown in the table below.

Time t (mins)	Frequency f
-10 ≤ t < -5	3
-5 ≤ t < 0	19
0 ≤ t < 5	32
5 ≤ t < 10
10 ≤ t < 20	53
20 ≤ t < 60	24

Write down, in the context of the question, what the time interval -10 ≤ t < -5  represents.

Hattie happens to notice that on three of the days she was recorded as being 40 minutes late for school, she had actually arrived 40 minutes early.  Calculate the corrected estimate for the mean amount of time she was late for school last year.

Roger has been looking at some data on the daily mean air temperature, t, in two different locations, Perth and Jacksonville, taken from the large data set.  All the data is taken from the month of July in 2015.

	n
Location A	31	836.3	22593.0
Location B	31			13.3	2.167

Unfortunately, some of the information has been lost and Roger does not know which data is for which location.