An investigation into the heights and weights of students attending Mayfield High School.

By looking at my graph I can see that the line of best fit is sloping upwards to the right, this tells me it has a positive gradient, therefore this graphs shows positive correlation, although as quite a lot of the points are spread around the line of best fit instead of being very close to it or on it this shows the positive correlation is weak instead of strong as I predicted.

Although I was wrong in my prediction the graph still shows support for my statement.

I will now work out the gradient of the line of best fit, by doing this it will tell me how much taller a student grows per kilogram of weight.

In order to do this I will use the formula: Gradient =

I will need to find two points which are on the line.

The points I have chosen are (48, 1.60) and (60, 1.68). I have chosen these points as they are on the line of best fit and therefore will make the gradient I work out more accurate.

The gradient = = 0.006666

= 0.007 - to one significant figure

This shows me that if a student's weight increases by one kilogram their height increases by 0.007m = 7mm.

I will now work out the equation of the line by applying the basic formula:

y = mx + c that I learnt about earlier in the maths' course.

In this case I will use h = mw + c, h will be standing for height and w standing for the weight, m is the gradient of the line and so m = 0.007

I know if h = 1.68, w = 60 so 1.68 = 0.007×60 + c

= 0.42 + c

This means c = 1.68 - 0.42 = 1.26 so the equation is h = 0.007w + 1.26

As a check I will substitute the other point, (48, 1.60) into the equation.

When w = 48, h = 0.007×48 + 1.26

= 0.336 + 1.26

= 1.596 m

The answer is close to what it should be but not quite right. This is probably because I rounded the gradient up to 0.007 or that the points are not quite on the line.

The number at the end of the equation, 1.26, is called the intercept and is the height value at the point where the line crosses the height axis. This would mean that a student whose height was 1.26 metres would actually weigh nothing at all. This is clearly not the case and I now remember that any line of best fit is only valid for the range of points plotted. I also think that in reality a curve of best fit would be better as height is only one dimensional whereas weight, being based on mass and volume, is three dimensional. Such a curve would probably indicate that a student of no weight would have no height as well. From what I have worked out I think that I have shown some support for my statement.

However I am now wondering if there is a difference in how much taller a boy grows for each increase of one kilogram when compared to a girl. So I am going to test the statement:

"Boys grow taller per kilogram than girls do"

To do this I shall draw separate scatter graphs of height against weight for the boys and the girls. I shall use the computer to do this and to show the lines of best fit with their equations. I am hoping that they will still show positive correlation but that the boys' line of best fit will be steeper than the girls'. This will be shown by the gradient of the boys' line being a higher value than the gradient of the girls' line.

The graphs are shown below.

The graphs do show positive correlation has indicated by both lines sloping upwards. That of the boys' seems to be a little stronger than the girls as the points are not spread as far from the line. The gradient of the boys' line is higher than the girls' so their line is steeper. The gradient of the boys' line suggests that the average boy grows 7.8mm per kg whereas the girls' only grow 5.7mm per kg. All of this, points to good support for my statement.

I am now going to compare the heights of boys and girls of equal weight to see if it gives any more evidence to support my statement. I shall choose a boy and a girl at 42kg and a boy and a girl at 64kg. I hope to show that the increase in the boys' heights is more than the girls' although the increase in the weights is the same. I will show the heights by drawing arrows on my graphs.

From this I saw that at 42kg the girl is 1.58m tall and the boy is 1.56m tall. This makes the girl taller than the boy by 2 cm, which surprised me as I thought it would be the other way round. At 64kg the boy is 1.72m tall and the girl 1.695 m tall. So now the boy is taller by 2.5 cm. This means that the boy has grown more than the girl, 4.5cm, although their weights increased by the same amount. I think this is very good support for my statement.

Since the boys grow more per kilogram than the girls I think the boys will be taller than girls in general. So for my next part of my work I will test the statement:

"Boys grow taller than girls as a general rule"

To test the statement I will put all of the boys' heights and all of the girls' heights into a back to back stem and leaf table. This will mean I can see all of the heights and compare the distributions. I hope it will show that the range of the boys' heights goes to taller values than that of the girls', though they will start at about the same height. I will also work out the quartiles and medians to show in the diagram.

I shall use the formula Lower quartile = height once all of the heights have been put into ascending numerical order. "n" stands for the amount of data which is 75 for both boys and girls.

Hence the lower quartile is = 19th height. The median will be the 19×2 = 38th height and the upper quartile the 19×3 = 57th height.

On my stem and leaf diagram I shall format two columns at each end to show the frequency (f) and the cumulative frequency (C.F.). The cumulative frequency will give me a useful check to see that I have included all the data as the final entry should be 75. It will also be helpful when locating the quartiles and median; I will put numbers at the bottom of the diagram to help me with this as well.

Later I will use the information from the stem and leaf diagram to draw whisker and box plots as these will make any differences between the boys' and girls' heights clearer,

I will now put the stem and leaf diagram below.

I can see that the distributions of both the boys' and the girls' heights are similar. Both have five heights between 1.40m and 1.49m but the boys have more heights above 1.80m than the girls' do. Indeed the boys' heights almost reach 1.90m whereas the girls' only just reach 1.80m. I think this is good support for my statement.

Now I will look at the box plots.

Once I had drawn my box plot I used the rule "whisker length < 1.5x the inter-quartile range, to check that I had no outliers in my data.

For the boys this meant there whiskers had to be shorter than 1.5 × 0.18 = 0.27m or 27cm.

For the girls it is 1.5 × 0.15 = 0.225m or 22.5cm.

Fortunately all of the whiskers were alright so I do not have any outliers.

Looking at the five key values of shortest height, lower quartile, median, upper quartile and tallest height; all but the shortest height are bigger in the boys' plot than the girls'. This is already support for my statement. Also the boys' range is more than that of the girls' indicating there is a greater difference in the boys' heights. This is also the case when looking at the inter-quartile range. I can see that the boys' heights do go on a further 9cm than the girls' heights thought they roughly start at the same height. This shows that the boys are taller. Also the medians are nearly in the middle of the inter-quartile range indicating a fairly even distribution for both sets of heights, however the girls' median is closer to the lower quartile than the upper quartile indicating that the girls' taller heights are slightly more spread out than the shorter ones. The girls' upper quartile is 4cm less than the boys' showing that more than a quarter of the boys are taller than three quarters of the girls.

I think I have shown enough evidence to support my statement which I now believe to be correct.

Earlier, when comparing heights of students of equal weight, I commented on being surprised by the girl being taller at 42kg. I think another way I could think of this is that the light students are in the lower years such as year 7 or possibly year 8. So it may be that when students start at secondary school the girls are taller than the boys. Then as they move up through the years the boys grow to be taller and for some reason the majority of the girls stop growing. Therefore for my next step I shall test the statement:

"In year 7 the girls are taller than the boys but by year 11 the boys have grown taller".

To do this I will put the heights of both boys and girls into back to back stem and leaf diagrams for each year. This means I will need to work out the five key values for each distribution. As there are 15 boys and 15 girls in each year group the lower quartile will be the 4th height, the median the 8th and the upper quartile the 12th. After drawing the diagrams and looking at the distributions I will use the information to draw whisker and box plots.

I will use a spread sheet to do this and use formulae in it to work out the inter quartile ranges and the ranges. I will also show a check that the lengths of the whiskers are less than the maximum length using the same rule as before.

From the spread sheet I can see that all the lengths of the whiskers are less than the maximum length allowable so my data does not contain any outliers when grouped in individual year groups. Also the lower quartiles and upper quartiles in years 7 & 8 are identical. This means that the inter quartile ranges will be the same which can also be seen in the above spread sheet.

I will now draw the stem & leaf diagrams and show them on the next page.

In year 7 the boys are taller than the girls at the beginning and end of the distributions. This, unfortunately, contradicts my statement and suggests that boys are 2 cm taller. However there is very little in it and the girls have more of the taller heights in the middle of the distribution. In fact both distributions are the same in terms of the number of heights in each part of the range.

In year 8 the boys have the shortest and tallest student but in between these, it is the girls who have the taller students.

So in these two years I think that although the boys have the tallest student there is evidence to support the statement that the girls are as tall as the boys if not taller.

In year 9 the girls still seem to be competing with the boys at both ends of the distribution but in between the boys have more students at the middle heights. I think that it is about this time that the boys start to pull away from the girls in terms of height.

In year 10 there are several more taller boys than girls so in this year I think it is safe to say that the boys are taller than the girls.

In year 11 the girls seem to have shrunk as their heights are now smaller than they were in year 10. This time the boys have several more students at heights greater than those of the girls.

I will now use the information from the table to draw box plots of the heights in individual years to see if there is any other support for my statement.

In years 7 and 8 I can clearly see that the lower quartiles, upper quartiles and inter quartile ranges are the same for both the boys and the girls. The median of the boys is greater than that of the girls in both year groups which may suggest that the boys are taller. I do think it shows that the girls are as tall as the boys generally and it may be due to the sample size and actual values in the sample that the girls have not been shown to be taller as they are in the scatter graphs.

The whiskers of the boys' heights in year 8 are long but within limits where as those of the girls are very short. This suggests that there is little difference between the girls' heights at this point whereas the boys show a much wider spread. This may indicate that some boys are beginning a growth spurt where others have not started. In fact the boys now have a student who is considerably shorter than the shortest boy in year 7.

The boys' median is more than that of the girls so there is a suggestion that they are starting to grow taller than the girls.

In year 9 all five key values are greater on the boys' plot than on the girls'. This shows that the boys are growing taller than the girls. The boys' inter quartile range is longer than it was in year 8 showing a greater dispersion in their heights and possibly reflecting the fact that they are beginning to grow taller. However the lower quartile is still the same value and the whiskers are short which again may indicate a growth spurt amongst the boys. On the other hand the boys' median is less than it was in year 8 which may support the idea that the boys are just beginning to start a growth spurt. The girls' inter quartile range is the same as it was in year 8 showing the central spread of their heights remains the same though the quartiles are lower than they are in year 8, which is surprising.

In year 10 the boys' key values are a lot more than the girls' with the exception of the median which is close to the lower quartile for the boys but the upper quartile in the girls' distribution. This suggests that the taller girls are similar in height but that the taller boys' heights are wide spread. This suggests that some boys may grow very tall which is something I have observed in real life. The boys' upper quartile is 7 cm more than that of the girls' and their median is only 2 cm less than the girls' upper quartile. This shows that nearly half of the boys are taller than three quarters of the girls. Even more convincing is the fact that the boys' upper quartile is more than the girls' tallest height so a quarter of the boys in this year group is taller than all of the girls. Assuming of course that the sample is representative of the year group though that is true of all the other comments as well.

However I do think I am very safe in concluding that at this stage the boys are taller than the girls.

In year 11 something weird seems to have happened as although all of the boys' key values are more than the girls' with the exception of the shortest heights which are the same. The key values are all less than they are in year 10. The plots still indicate that the boys are taller than the girls but it is very strange that the students appear to have shrunk from year 10 to year 11. I am wondering if when the data was put in to the database the person doing it confused the two year groups. If this is the case then this person must be incompetent considering all the other errors in the data. Maybe there was more than one person inputting the data but the other idea I have is that Edexcel deliberately swapped the year groups data to confuse the user.

However I think there is clear evidence that the boys do grow taller than the girls but the evidence that girls are taller in year 7 is inconclusive. The plots clearly show that both genders grow taller as they get older. The median height, which is unaffected by extreme values, is always greater for the boys than the girls indicating that the boys are taller in general. Also in years 10 and 11 the boys' medians are at the same height, weird.

I think I have shown that my statement is at least half true though I believe that it would have been better if I had been able to follow the same group of boys and girls through all the 5 years of secondary education. Perhaps Edexcel could consider providing this data as they have had at least five years to collect the data given that the current data is based on an actual school in the North of England.

Grace Ball.

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