Processing
In order to obtain my sample, I did some stratified random sampling. This means that I took 50 boys and 50 girls form year 7, but used the random numbers generator on my calculator to pick which students would be selected for my sample. I did this so that the information I gathered would not be bias in any way.
The first thing I did was to find out the average mark from the key stage 2 results for boys and girls. I was left with the following results:
Mean results - boys KS2.
English
3 + 5 + 4 + 4 + 5 + 3 + 4 + 5 + 4 + 4 + 5 + 4 + 5 + 5 + 4 + 4 + 4 + 2 + 5 + 3 + 4 + 4 + 4 + 5 + 3 + 4 + 4 + 5 + 4 + 4 + 4 + 4 + 4 + 5 + 4 + 4 + 5 + 4 + 4 + 3 + 4 + 5 + 4 + 5 + 5 + 4 + 5 + 3 + 3 + 3 = 201
201 = 4.02
50
Mean for boys' English results is level 4
Maths
4 + 3 + 5 + 5 + 4 + 3 + 4 + 4 + 3 + 4 + 5 + 4 + 5 + 5 + 3 + 5 + 4 + 3 + 5 + 4 + 5 + 3 + 5 + 4 + 4 + 4 + 5 + 5 + 4 + 4 + 4 + 4 + 4 + 5 + 4 + 3 + 5 + 3 + 4 + 3 + 4 + 5 + 5 + 5 + 5 + 4 + 5 + 4 + 4 + 3 = 208
208 = 4.16
50
Mean for boys' Maths results is level 4
Science
4 + 5 + 3 + 5 + 5 + 4 + 4 + 4 + 4 + 4 + 5 + 4 + 5 + 5 + 4 + 4 + 4 + 3 + 5 + 3 + 4 + 3 + 5 + 5 + 3 + 3 + 5 + 4 + 4 + 4 + 4 + 4 + 4 + 5 + 4 + 4 + 5 + 4 + 5 + 3 + 4 + 5 + 5 + 5 + 4 + 4 + 4 + 4 + 4 + 3 = 208
208 = 4.16
50
Mean for boys' science results is level 4
Cumulative frequency tables for the Boys' Key stage 2 Results
English
Maths
Science
For the boys' Key stage 2 results, you can see from the curve that all three subjects are very close in the results that the students achieved. All three subjects had the same average level for the boys.
As you can see from the cumulative frequency table of results, most of the boys achieved a level 4 in all three subjects, which shows that the majority of boys do not excel in one particular subject.
The following table shows the chart with which I worked out class widths and frequencies for the histogram that shows the results for the boys' IQ.
I decided to use these class intervals for the construction of this histogram because they are most appropriate. The first class width started at 70 because from the information that I had gathered it did not show that any boys were achieving below this mark in their IQ. As there are not many people who were getting between 70 and 80 for their IQ results, I decided to make the interval slightly wider to 70 - 90 points. I used the next interval because there were a few people who would be put into this interval. I thought it best to have one overall interval that held the majority of students. I chose a small interval again for the next two because it seemed that they did not contain many students.
The average IQ result for the boys was 99.
The next thing I did was to find the average for the Girls' Key stage 2 results. Here is what I found:
English
5 + 5 + 5 + 5 + 5 + 5 + 4 + 4 + 4 + 4 + 5 + 4 + 4 + 5 + 4 + 4 + 3 + 3 + 4 + 3 + 5 + 5 + 4 + 5 + 4 + 4 + 3 + 3 + 4 + 4 + 4 + 4 + 5 + 4 + 5 + 3 + 5 + 4 + 4 + 5 + 3 + 5 + 5 + 5 + 5 + 4 + 5 + 5 + 5 + 4 = 215
215 = 4.3
50
Mean average girls' English result is level 4
Maths
6 + 4 + 5 + 5 + 4 + 5 + 4 + 4 + 4 + 4 + 5 + 4 + 3 + 5 + 4 + 4 + 3 + 3 + 4 + 3 + 5 + 5 + 4 + 5 + 4 + 3 + 3 + 3 + 4 + 4 + 4 + 4 + 4 + 5 + 5 + 3 + 4 + 4 + 4 + 5 + 3 + 5 + 4 + 5 + 4 + 5 + 5 + 5 + 5 + 4 = 210
209 = 4.2
50
Mean average girls' Maths result is level 4
Science
5 + 5 + 5 + 5 + 4 + 5 + 4 + 4 + 4 + 4 + 6 + 4 + 4 + 5 + 5 + 4 + 3 + 4 + 5 + 3 + 4 + 5 + 4 + 5 + 4 + 4 + 3 + 4 + 4 + 5 + 5 + 4 + 4 + 5 + 5 + 4 + 4 + 5 + 4 + 5 + 3 + 5 + 5 + 5 + 4 + 5 + 5 + 5 + 6 + 5 = 223
223 = 4.56
50
Mean average girls' Science results is level 5
However, although the mean averages for the girls' results say that the majority of girls are achieving level 4 in their key stage 2 results, if you compare this with the mode averages, then I get slightly different results.
The mode averages suggest that the majority of girls are achieving level 5 in their key stage 2 exams. (Excluding the maths results with which the average would still be level 4). The following tables show this:
Percentage and cumulative frequency tables for the girls' Key stage 2 Results
English
Maths
Science
For the girls' key stage 2 results, you can see by the cumulative frequency results on the graph that the marks are more spread apart, and there are more numbers of girls who are getting higher marks. This therefore shows that the girls are more able to excel in certain subjects than boys, who seem to be achieving the average mark in all subjects.
The following table shows the results that I gathered in order to complete a histogram to show the range of IQ results for the girls.
I used these class intervals for the same reasons as in the boys IQ histogram results. However, there is a difference in the last interval. This is because there Ire higher results for the girls than the boys, so in order to use the whole of my sample, I decided it would be best to widen the last interval slightly. The mean average for the Girls IQ results was 105 points.
I have done a data check on all the calculations that I have made but the accuracy of my data is not as accurate as it could be because most of my numbers have been rounded up to make it easier to present.
For the second investigation I found that there were 282 students in year 7. I wanted an equal proportion of girls and boys so I did the following calculations:
I then randomly chose 54 boys and 46 girls. I then made a graph on excel where Key Stage 2 result was on the vertical axis and IQ result on the horizontal axis.
Interpreting
The results that I gathered in general show that the girls from my sample are more intelligent that the boys. The girls are achieving higher key stage 2 results and higher IQ results. It is also true that the girls must put more effort into their studies in order for them to be achieving higher marks, and are therefore learning more rapidly because of this. I would like to stress that there were a few boys in my sample who did achieve above the average mark for their gender, but this was only a minority. The girls' majority of results proved to be higher than the boys. The anomalies could be because the pupil could have been ill or just did not try very hard for the test there are a number of reasons why there are anomalies within the data.
Therefore in conclusion, I would like to state that this does not always mean that girls are better than boys or vice versa, but that the majority of the girls would work harder to achieve good results than the boys.
From the histograms showing the IQ results, I can conclude that the girls are getting higher results than boys due to a higher area covered by the bar in the girl’s range 110-135 whilst the boys range only goes up to 120. The Histogram representing the boys IQ result shows that the results are grouped nearer the mean and the girls IQ results are more spread out. The cumulative frequency curves support my idea because the lower the curve the higher the result, the lower curves were always the girl’s results meaning that they are more intelligent than the boys. The girl’s frequency polygons are more to the right than the boy’s frequency polygons this shows that the girls are scoring higher marks and therefore are cleverer.
The scatter graph clearly shows a positive correlation this indicates that there is a relationship between the IQ results and the Key Stage 2 results. As the correlation is positive my theory that the IQ result and the Key Stage 2 results are directly proportional is right because as the IQ result increases the Key Stage 2 result increases. Although there are two anomalies this could be because the individual did not try hard or found it hard.
Hypothesis 2
I think that males have a greater body mass index than females. I predict that this will be true because males tend to be taller and therefore weigh more.
Plan
I plan to take a stratified sample of a 100 pupils from year 11, 50 of which will be girls and 50 boys. The body mass index can be worked out by simply dividing the weight by the height squared, the formula is shown below:
Body mass index = Weight
Height ²
I will then work out the standard deviation of the results and show the standard deviation on a histogram. The standard deviation is how far from the mean the data is spread. The formulae for the variance and standard deviation are given below. m means the mean of the data.
The standard deviation, s, is the square root of the variance.
What the formula means:
(1) xr - m means take each value in turn and subtract the mean from each value.
(2) (xr - m) ² means square each of the results obtained from step (1). This is to get rid of any minus signs.
(3) S (xr - m) ² means add up all of the results obtained from step (2).
(4) Divide step (3) by n, which is the number of numbers
(5) For the standard deviation, square root the answer to step (4).
Processing
Firstly I worked out the body mass index for my sample, and then I worked out the mean body mass index for females and males separately. Subsequently I step by step worked out the standard deviation, starting with the variance sq. On the following pages you can see clearly how I worked out the standard deviation.
Interpretation
When comparing the two histograms for both female and male, it is clear that boys are generally heavier than the girls. The mean body mass index is 20.47 for the boys and 18.45 for the girls; these values also lie in the modal range. I have calculated the standard deviation for boys is 4.38 and for girls it is 4.26. This indicates that the body mass index for girls (which is less than boys) in more concentrated around the mean. Overall the graphs are more nearer to a normal distribution i.e. bell shaped with the highest point at the mean. There are always some anomalous results which are generally dispersed away from the mean. Using the standard deviation we can get a more accurate range of body mass index for males is between 14.8 and 23.56, for females are between 14.19 and 22.71. These ranges can be useful when determining if someone is underweight or overweight. On average a person’s body mass would fall in the above ranges depending on their size. Both graphs show a tight distribution of results where most of the values are within a narrow range either side of the mean.
My results are not completely accurate due to the fact that I have rounded most of my figure to give an integer this means that my results are slightly wrong. Also when sampling I could have been biased and chosen the data to suit me this would mean that my results are not as precise as they should be. There are many factors which could slightly jeopardise my results such as there could have been a type error the list could go on.