The large data set Statistics

Mark scheme (whole question; this part is marked with the orange bar)

Examiner's comment

Part (c) proved more challenging for most, demonstrating that students are still not familiar with the large data set and how to apply it to a scenario. Of the students who were able to successfully identify the number of moderate days, most were able to make use of the upper quartile to arrive at a correct estimate of 5 or 6 days. A small number of students calculated three quarters of the days instead of one quarter.

Common mistakes

• needs LDS knowledge that Perth 2015 has 184 days (May to October) so 184 − 161 = 23 Moderate days
• 1028 is the upper quartile for Moderate days so a quarter: 23 × 0.25 = 5.75, so 5 or 6 days
• some used three quarters

Mark scheme (whole question; this part is marked with the orange bar)

Examiner's comment

In part (b) most identified that “trace” or “tr” would need attention, but some did not link this to the rainfall variable. In part (c) (i) there were many correct answers and some who

Common mistakes

• daily total rainfall may contain "tr" (trace) entries that need a value before calculating
• must link it to rainfall

Mark scheme (whole question; this part is marked with the orange bar)

Examiner's comment

to the rainfall variable. In part (c) (i) there were many correct answers and some who identified mm or hours but not in a correct combination: answers such as mm2 or even -were seen, whilst others appeared not to be that familiar with the large data set and 1 hour mm used other incorrect units such as oktas, lumens or Hpa. Most responses gave a suitable contextual response to part (ii) but some simply said that 0.714 was the minimum rainfall rather than the (average) rainfall when there was no sun. The majority of responses recognised the inconsistency between Kay’s teacher’s claim and Kay’s data and many cited “positive correlation” for Kay’s data as their reason. A few opted to substitute two different values into the equation of the line to demonstrate their point. A small minority believed the teacher’s claim to be true and they were perhaps relying on their knowledge of weather conditions rather than interpreting the data Kay had collected.

Common mistakes

• units of the gradient mm per hour (rainfall mm, sunshine hours), not oktas or hPa
• intercept = (average) rainfall when there is no sunshine, not "minimum rainfall"

Mark scheme (whole question; this part is marked with the orange bar)

Examiner's comment

learners and very few blank responses were seen and parts (e) and (f) provided an opportunity for differentiation. gave a correct conclusion in context. Good attempts to part (f) were rare and blank responses not uncommon. Many answers did not give full and precise definitions and others lacked clarity so that they could have been interpreted as either the sample or a population. For part (i) the information was given in the question but few were able to extract the relevant information: the variables (sun and rain) for May to October (or from the large data set) from 2015. Part (ii) produced a slightly better response with successful comments such as “sun and rain from all of 2015” or “all the variables in the large data set” or “from all across the UK” being seen. Common mistakes often involved describing a population for which the sample was not a subset such as using months from April to November or the year 2016.

Common mistakes

• sample = rainfall and sunshine for Leeming May to October 2015 (all the LDS days)
• population must contain the sample, e.g. all of 2015 or other years, not April to November or 2016 only

Mark scheme (whole question; this part is marked with the orange bar)

Examiner's comment

This question was generally not well attempted demonstrating that students are still not completely familiar with the large data set. In part (a) many students appeared to be aware of Camborne's location in the South-West of the UK but could not relate this to the prevailing winds in that area. Some students were able to guess that A represented East but were unable to provide a reason. Many students did not appreciate that wind direction is the direction that wind blows from, rather than to.

Common mistakes

• A is East: the least common direction, since prevailing UK/Camborne winds come from the south-west (west/south)
• wind direction is where the wind blows from, not to
• a reason is needed

Mark scheme (whole question; this part is marked with the orange bar)

Examiner's comment

This question was generally not well attempted demonstrating that students are still not In part (b), many students appreciated that the value was an outlier and should be excluded but were unable to score the marks here without justification. Those who understood that the maximum that the angle could be 360, tended to score well here. Some students also thought that 999 represented a frequency rather than an angle.

Common mistakes

• wind direction runs from 0 to 360 degrees so 999 is not a valid value (it is an error/missing-data code, not an outlier or a frequency)
• remove/ignore it
• a reason using 360 is needed before the action

Mark scheme (whole question; this part is marked with the orange bar)

Examiner's comment

It is encouraging to see most students now attempting the question on the large data set and a growing familiarity with it. In part (a) many wrote down rainfall or windspeed but other quantitative variables such as hours of sunshine or maximum wind gust were sometimes seen but these are not variables available for overseas locations. Some students gave answers like “trace” or “outlier” which have appeared in previous papers but had no relevance to this question. Part (b) was a

Common mistakes

• rainfall (or mean windspeed)
• hours of sunshine and gusts are not recorded for overseas locations
• "trace" or "outlier" irrelevant here

Mark scheme (whole question; this part is marked with the orange bar)

Examiner's comment

It is encouraging to see most students now attempting the question on the large data set and a growing familiarity with it. The final part of the question required students to know that Perth on the large data set was in the southern hemisphere and therefore the warmer months would be during our autumn and winter. They were told that the mean air temperature for the month in question was 19.4 and needed to observe that this was larger than the overall mean of 15.2 for the 6 months of the large data set. Using their knowledge that the large data set only covers the months from May to October they could then infer that the month in question was October. Many were able to identify that the month to be identified was a warmer month (and scored the first mark) but then suggested an answer such as November or December which of course are not available in the large data set. It was sometimes difficult to know whether a candidate was using “summer” to simply refer to the months from June to August or referring to the warmest months of the year in Perth, often an example month was given which helped clarify what the student meant.

Common mistakes

• 19.4 is above the overall mean 15.2 so a warmer month
• Perth is in the southern hemisphere and the data set covers May–October, so October
• November/December are not in the data set

Mark scheme (whole question; this part is marked with the orange bar)

Examiner's comment

but due to parts (c) and (d) requiring familiarity with the large data set, only a small minority of students scored full marks here. Most students successfully answered part (a) with a correct interpretation of a lack of standard in parts (c) and (d) was poor. Most students guessed an answer to part (c) but very few were able to give an answer in the correct range. In part (d), the students often described

Common mistakes

• interpolation needs a pressure within the LDS range for Beijing 2015, 990 to 1040 hPa
• most students guessed

Mark scheme (whole question; this part is marked with the orange bar)

Examiner's comment

but due to parts (c) and (d) requiring familiarity with the large data set, only a small minority of students scored full marks here. Most students successfully answered part (a) with a correct interpretation of a lack of standard in parts (c) and (d) was poor. Most students guessed an answer to part (c) but very few were able to give an answer in the correct range. In part (d), the students often described the location of Beijing as a factor or repeated an answer given in previous examination series rather than comment on the Beaufort Conversion being qualitative data. It was clear the students who had studied the large data set as their answers had a concise explanation.

Common mistakes

• Daily Mean Wind Speed (Beaufort Conversion) is qualitative/categorical (given in words)
• "not continuous" alone not accepted
• location of Beijing is not a reason

Mark scheme (whole question; this part is marked with the orange bar)

Examiner's comment

Once again the large data set aspects of this question caused problems for some students but most scored some marks here. In part (a) a few responses simply quoted standard textbook answers about cleaning data by removing outliers and others were distracted by the class intervals for the data. Many though did realise that the “tr” values needed attention but a number didn’t seem to realise that trace entries had a value between 0 and 0.05 and therefore removing them would have a big effect on the calculations Ben wanted to carry out. Those that did appreciate the value of the “tr” usually suggested a suitable numerical value (usually 0) to replace them with and scored both marks.

Common mistakes

• tr (trace) means less than 0.05 mm: replace with a numerical value such as 0 or 0.025, do not remove
• removing outliers or equalising class widths is not cleaning here

Mark scheme (whole question; this part is marked with the orange bar)

Examiner's comment

Once again the large data set aspects of this question caused problems for some students but most scored some marks here. the large data set and scoring some marks in part (c). In (i) we were simply looking for an awareness that because the large data set only had data for May to October this would not be representative of the whole population. Some demonstrated a lack of understanding of the principle of using a sample to make inferences by stating that 184 days was not a sufficiently large sample to represent the weather for the whole year. In this case the problem was not the sample size but the fact that the sample was not representative. In (ii) even some of those who had failed to mention May to October in (i) were able to score here by stating that because the missing months were in winter and they expected more rainfall then, the actual mean would be higher. The great storm of October 1987 featured in many incorrect answers to part (c).

Common mistakes

• the large data set only covers May to October, so not representative
• "184 days is too small a sample" scores B0
• winter months are wetter so (b)(i) is an underestimate
• the October 1987 storm is not the point

Mark scheme (whole question; this part is marked with the orange bar)

Examiner's comment

Although a few students simply left this question blank most had a go though only about a quarter scored either of the marks. The median was the most successfully answered with 1000 being a common acceptable value. Selecting a suitable value for the range was more challenging with most responses being far too large. Although the specification requires students to be “familiar” with the large data set, for this sitting it is understandable that many would not have had as much time for familiarisation as would be desirable.

Common mistakes

• LDS daily mean pressure is around 1000 to 1020 hPa (accept 990 to 1030), not 1200

Mark scheme (whole question; this part is marked with the orange bar)

Examiner's comment

Although a few students simply left this question blank most had a go though only about a quarter scored either of the marks. The median was the most successfully answered with 1000 being a common acceptable value. Selecting a suitable value for the range was more challenging with most responses being far too large. Although the specification requires students to be “familiar” with the large data set, for this sitting it is understandable that many would not have had as much time for familiarisation as would be desirable.

Common mistakes

• range of daily mean pressure in a month is about 10 to 50 hPa
• most answers far too large

Mark scheme (whole question; this part is marked with the orange bar)

Examiner's comment

Part (a) tested knowledge of the data set. Some students were able to identify tr or trace as the required answer, but many displayed their lack of knowledge of the data set, with n/a being a typical wrong answer, or 0 or 0.025. Many students gave no answer of any sort and there were a few nonsensical or confused answers such as explaining how the rainfall might be collected (a particular favourite being ‘with a bucket and ruler’) or how it might be recorded (in an excel spreadsheet).

Common mistakes

• tr (trace)
• n/a, 0 or 0.025 are wrong

Mark scheme (whole question; this part is marked with the orange bar)

Examiner's comment

Part (c) proved to be more problematic. Many students lacked knowledge of the LDS and did not know the relative geographical locations of Camborne and Leuchars within the UK. As such, a significant number of students made the incorrect assumption that Leuchars is in the South and Camborne is in the North. However, when students were able to locate the relative positions of the two places correctly, they were generally able to draw the correct conclusion, resulting from comparing the mean rainfall between the two places. The main errors included • not making reference to the relative positions of the two locations in their answer e.g. stating that Camborne was in the South with no reference to the relative position of Leuchars, • not making any statements about either of the geographical locations at all and only compared means, with responses such as: no, as the mean is less in Leuchars than in Camborne. • not drawing the correct conclusion after correctly comparing locations and means • not stating any conclusion. • incorrectly comparing standard deviations rather than means.

Common mistakes

• Leuchars is north of Camborne (many got this the wrong way round)
• Leuchars has the smaller mean so no evidence for the belief
• compare means, not standard deviations

Mark scheme (whole question; this part is marked with the orange bar)

Examiner's comment

Although some showed a genuine familiarity with the large data set many seemed totally unfamiliar with it and subsequently could not make progress in part (b). For part (b) the students needed to be familiar with the qualitative variables in the large data set (wind speed and cardinal direction were the only choices) and then select an appropriate variable in the knowledge that there were only 3 outcomes. Those who were familiar with the data set would then know that the majority of wind speeds are “light”. Aside from the large data set, most correctly identified the sampling technique in part (a) and some drew a sensible

Common mistakes

• qualitative variables in the LDS are wind speed (Beaufort descriptions) and cardinal wind direction
• only 3 outcomes means daily mean wind speed
• modal outcome A is "light"
• "wind" or "wind direction" not accepted

Mark scheme (whole question; this part is marked with the orange bar)

Examiner's comment

Although this question involved familiarisation with the large data set nearly 70% scored 3 or more marks here. Many students knew the units used for Daily Mean Pressure (or wrote “Pascals” which was allowed).

Common mistakes

• Daily Mean Pressure is in hectopascals (hPa)
• kPa or Pa alone not accepted

Mark scheme (whole question; this part is marked with the orange bar)

Examiner's comment

Although this question involved familiarisation with the large data set nearly 70% scored 3 or more marks here. Many students knew the units used for Daily Mean Pressure (or wrote “Pascals” which was allowed). reaching the correct value of 12.1 and this just lost the first method mark for this part. The final part proved challenging. There was evidence that students knew the relative locations of the 3 sites and often they deduced that it was high pressure and therefore the winds circulated clockwise. The major problem though was that students did not seem to appreciate that Cardinal Wind Direction in the large data set gives the direction from which the wind comes. This led to an incorrect allocation of directions in the table, some students were clearly confused by this and felt that the circulation must be anticlockwise and then a correct allocation of wind directions could, of course, be deduced but this lost the first mark. A small percentage of students though did achieve full marks for this question.

Common mistakes

• 1028–1029 is about mean + s.d., so high pressure and clockwise
• locations north to south Leuchars, Heathrow, Hurn
• Cardinal Wind Direction is the direction the wind blows FROM: Heathrow NE, Hurn E, Leuchars W

Mark scheme (whole question; this part is marked with the orange bar)

Examiner's comment

It was clear from the responses seen here that many candidates have little or no experience with the large data set. In part (a) very few students knew that the figures for Daily Mean Visibility are in decametres (Dm). In part (d) a minority of

Common mistakes

• visibility in the large data set is in decametres, so 0 to the nearest 100 dam means 0 to 500 m (or under 50 dam)
• very few knew the units

Mark scheme (whole question; this part is marked with the orange bar)

Examiner's comment

It was clear from the responses seen here that many candidates have little or no figures for Daily Mean Visibility are in decametres (Dm). In part (d) a minority of students gave the correct variable hours of sunshine (just Sunshine was accepted) as the variable.

Common mistakes

• (hours of) sunshine
• not cloud cover (values above 8 oktas), not wind speed (values not integers), not mean temperature (near-zero unlikely in June)
• give one variable only

Mark scheme (whole question; this part is marked with the orange bar)

Examiner's comment

though this was not required here. In part (b) it was clear that more students were familiar with the large data set. Many identified “rainfall” as the variable on the y-axis and they usually gave the correct units although some thought it should be mm3. The most common incorrect answer here was “cloud cover” but, of course, this variable is not available on the large data set for Perth. Part (c) was a standard hypothesis test for

Common mistakes

• Perth has no cloud cover
• rainfall (mm) or pressure (hPa) – the only other Perth variable, wind speed, is positively correlated with temperature
• mm³ is wrong

Mark scheme (whole question; this part is marked with the orange bar)

Examiner's comment

In part (a) most candidates referred to data that was n/a, anomalies or outliers with only a small proportion appreciating that the data set contained trace values for rainfall which needed replacing with numerical values.

Common mistakes

• LDS: rainfall includes "tr" (trace, less than 0.05 mm) which must be converted to a number
• n/a, missing data or outliers are not the answer

Mark scheme (whole question; this part is marked with the orange bar)

Examiner's comment

The underlying assumption in part (d)(i) was generally identified but not always with sufficient clarity to gain the mark. Common errors were: ‘data uniformly distributed’ (failing to mention within each class); the average of each class is the midpoint (failing to specify the mean of each class); most of the data is located at the midpoint (failing to identify that the midpoint is assumed to represent all of the data). It was clear from the responses seen here that many candidates have little or no experience with the large data set. In (d)(ii) quite a number referred to the data being based on weather which is unpredictable and that therefore the previous assumption was unrealistic. Good explanations involving the distribution of the bottom class were rare with very few making reference to the majority of data being either 0 or close to 0. Little progress was made in (d)(iii) as many mentioned irrelevancies such as the great storm or the fact that it was during the summer or that the database did not cover a full year. Many did not realise that they were comparing an estimated mean with an actual mean for the same time span and said the actual mean would be higher because the winter months were excluded.

Common mistakes

• (i) values are spread evenly within each class (not "data uniformly distributed")
• (ii) LDS: most days in the first class have 0 or trace rainfall
• (iii) so the actual mean is smaller than the estimate
• comparing with winter months or storms is irrelevant

Mark scheme (whole question; this part is marked with the orange bar)

Examiner's comment

candidates. Parts (b) and (e) were a problem for those less familiar with the large data set. part (b) that many were not aware that for all locations the data is for the months of May to October only. Successful candidates often explained that Beijing was in the northern hemisphere and so low outliers for temperature were likely to be for winter, and so October. Some candidates perhaps did not read the question carefully enough and stated two months. S Most candidates scored well in part (c), usually using the given formula. A small xx

Common mistakes

• LDS covers May to October only
• Beijing is in the northern hemisphere so the coldest month is October
• give one month only

Mark scheme (whole question; this part is marked with the orange bar)

Examiner's comment

candidates. Parts (b) and (e) were a problem for those less familiar with the large data set. Part (e) was very poorly done with only a handful of candidates scoring marks here. Depth of familiarity was lacking by many candidates. Some misunderstood the question entirely and cited the conditions for a Normal being a suitable approximation to a Binomial. Others thought this part was related to the variable under study and suggested the Normal was not suitable for modelling air temperature for a variety of reasons including the skew on the box plot. Those who did suggest two other variables from the LDS rarely gave acceptable reasons for their choice; some gave no reasons at all even when their choice was a suitable one. The use of the Beaufort scale for wind speed was mentioned but it was wrongly said to be discrete with only a few saying that it was non-numeric. Wind direction which was non-numeric was quite often seen but was not an acceptable answer. Some other wrong reasons were: the data is non- uniform, the values are too variable; it is not possible to have values below 0. The most commonly suggested variable was cloud cover, presumably because it was used as the basis for Q4 and possibly because it appeared in last summer’s 8MA02

Common mistakes

• rainfall (skewed, many days of 0), wind speed Beaufort (non-numeric), pressure, date (uniform)
• "discrete" or "not continuous" not accepted
• wind direction not accepted
• normal-approximation-to-binomial conditions are irrelevant

Mark scheme (whole question; this part is marked with the orange bar)

Examiner's comment

The new specification requires students to be familiar with the large data set and the first part of the question addressed this issue. In part (a) the students needed to realise that n/a in the large data set means that the data for that point was not available and this means that there are only 18 (not 31) pieces of data available. Another skill students are required to have under the new specification is to be able to calculate standard statistics using their calculators. In this case, we expected students to enter the data into their calculator and write down the mean (part (a)) and standard deviation (part

Common mistakes

• n/a means data not available, so n = 18 not 31 (treating n/a as 0 gives 5.94)
• mean 184/18 = 10.2
• use the calculator's statistics mode

Mark scheme (whole question; this part is marked with the orange bar)

Examiner's comment

(b)). It was clear that many students were not familiar with this requirement and we often saw several lines of working to derive these results which unfortunately under the new specification cannot receive any marks. A small number of students knew that wind speed was measured in knots but many did not know this.

Common mistakes

• σ = 3.17 (s = 3.26 accepted) from n = 18
• units are knots (kn), which many did not know
• long hand derivations earn no extra credit

Mark scheme (whole question; this part is marked with the orange bar)

Examiner's comment

In part (c) there were many good attempts and most students gave a correct response here and in (d)(i) most knew that the * represented an outlier. Part (ii) was very challenging and required some careful thought and inference. Some confused mean with median and assumed that C and D were the 2nd and 3rd box plots in the list since they had the same medians. The intention was that they should identify E as the 5th box plot and A as the 2nd (it has a low mean and standard deviation and clearly has the smallest range on the box plots). Looking at the other 3 they should notice that the range is large and the outlier too would suggest a large standard deviation. The box plot Y has the smallest median of these 3 so a lower mean is suggested which points to B. We were looking for some explanation using correct terminology so we expected references to the mean and median and the standard deviation and a suitable measure of spread from the box plots.

Common mistakes

• October (accept September): Camborne is windier in autumn, October 1987 was the month of the hurricane, or it is the latest month in the data

Mark scheme (whole question; this part is marked with the orange bar)

Examiner's comment

This question was supposed to be a gentle starter, making use of the student’s knowledge of the large data set. Unfortunately many did not seem to be familiar with this and certainly were not aware that cloud cover was measure in oktas and subsequently there were a number who left the question blank. Those who did know about the cloud cover variable were able to give a discrete uniform distribution but many forgot to include 0. Those who did try and engage with the question could often score in

Common mistakes

• cloud cover is measured in oktas 0 to 8, so 9 values each with probability 1/9
• many forgot to include 0
• many unfamiliar with the variable and left it blank

Mark scheme (whole question; this part is marked with the orange bar)

Examiner's comment

This question was supposed to be a gentle starter, making use of the student’s knowledge of the large data set. Unfortunately many did not seem to be familiar with this and certainly were not aware that cloud cover was measure in oktas and subsequently there were a number who left the question blank. Those who did know about the cloud cover variable were able to give a discrete uniform distribution but many forgot to include 0. Those who did try and engage with the question could often score in

Common mistakes

• less than 50% means fewer than 4 oktas, so P(C < 4) = 4/9 (0.444)

Mark scheme (whole question; this part is marked with the orange bar)

Examiner's comment

part (c) but responses to part (d) were poor. This question required students to suggest a refinement to Helen’s model and most answers lacked the detail required for the mark.

Common mistakes

• just saying "not uniform" is B0
• refinement must use variation by month or location (e.g. a non-uniform distribution with probabilities from frequencies)
• a continuous model such as normal is B0
• most answers lacked detail