I will draw 2 histograms from the 2 grouped frequency tables for each year. The purpose of the histograms is to double check the data and to illustrate the dispersion of data between the 2 years. From the histogram I will find out the frequency of guesses within 20cm of the actual length. The year with the most guesses will be more accurate.
This is how I have achieved my first random sample:-
- Shift Ran# = 0.884
- 0.884 x 173 = 152.932
- round the number off to 153. therefore that is my first sample.
Year 7
Whilst I was generating numbers, I came across a few repeats which I ignored and carried on. Also, as you can see on the table above, I came across an out-of-context number which I also replaced. This is because it is an extreme value which will distort my investigation. I will now construct a table for Year 11.
Year 11 random sample
I will now construct 2 grouped frequency tables – one for Year 7 and one for Year 11.
Year 7 grouped frequency table
To find the frequency density I used the following equation:
Frequency density = Frequency
Class Width
I will now do the same for Year 11:-
Year 11 Grouped frequency table
I used the same equation to find the frequency density.
I will now draw 2 histograms from my results.
I will construct 2 further tables to help me find an estimate of the mean and standard deviation.
Year 7
∑f = 50 ∑fx = 77.675 ∑fx2 = 125.501 (3dp)
Mean = ∑fx = 77.675 = 1.5535
∑f 50
S.D = ∑fx2 - ∑fx = 125.501 - 77.675
∑f ∑f 50 50
= 2.51002 - 2.41336225
= 0.09665775
= 0.310898295
= 0.311 (3dp)
Now that I know what the mean and standard deviation for Year 7 is, I will apply the same calculations to Year 11 to find their mean and standard deviation.
Year 11
∑f = 50 ∑fx = 79.60 ∑fx2 = 129.187 (3dp)
Mean = ∑fx = 79.60 = 1.592
∑f 50
S.D = ∑fx2 - ∑fx = 129.187 - 79.60
∑f ∑f 50 50
= 2.58374 - 2.534464
= 0.049276
= 0.221981981
= 0.222 (3dp)
Conclusion
Overall, my results conclude that Year 11 is better at estimating the length of an object than Year 7. Therefore this means that my hypothesis is right. I proved this from my investigation and results which show that:-
- Year 11’s mean is closer to the true value of the bamboo stick than Year 7’s. This proves my hypothesis is right because Year 11’s estimates were more accurate.
- Year 11’s standard deviation is lower than Year 7’s. This means my hypothesis is correct because the Year 11’s estimates are more condensed.
The histograms I have drawn show that a larger amount of the Year 11’s estimates are in the frequency group that contain the actual length of the stick than Year 7’s estimates. The Year 11’s estimates are more condensed and nearer to 1.58m.
From my histogram I will calculate the amount of guesses for each year that are within the range of 20cms either side of 1.58m.
Year 7: (35 x 0.12) + (85 x 0.08) + (85 x 0.12) + (30 x 0.08)
= 4.2 + 6.8 + 10.2 + 2.4
= 23.6 estimates were in 20cms either side of 1.58m
Year 11: (40 x 0.02) + (50 x 0.10) + (120 x 0.08) + (120 x 0.12) + (35 x 0.08)
= 0.8 + 5 + 9.6 + 14.4 + 2.8
= 32.6 estimates were in 20cms either side of 1.58m.
Te results conclude that there are more estimates within either side of 1.58m in the Y11 histogram than there are in the Y7. This means that there are more Y11 estimates within the 20cm range than Y7.
Evaluation
If I had a chance in the future to carry out this investigation again, there are some things I would change in order to improve it.
First of all I would use a larger sample of estimates to produce more accurate results and improve the reliability.
Secondly, I would be interested in comparing estimates of boys with girls and I could use different age groups such as compare young people with older people (OAPs for example). Also, I would not only compare Year 7 with Year 11, but compare other year groups as well.
Thirdly the data I was given is secondary, so it might have affected the results. I would collect my own primary data from our school.
Finally, for my statistical calculations I would not only find out the mean and standard deviation, but I would also find out the range and mode.
Overall I feel that the investigation went well and I am pleased with the results. There was a small problem when I was generating my random samples. One of the estimates that came up was an extreme value of 12.13 metres. I replaced this number with another one because if I had included it then my results would be distorted.
