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  • Level: GCSE
  • Subject: Maths
  • Word count: 6409

Data comparing year 7 and year 11 length estimates - How well can you estimate the length of a stick?

Extracts from this document...

Introduction

Year Seven

Gender

Length (m)

F

1.50

M

1.89

F

1.11

F

1.50

F

1.61

M

1.20

F

1.04

M

1.00

F

1.50

F

1.57

F

1.45

F

1.20

M

1.25

F

1.63

F

2.00

F

2.30

M

1.45

F

1.25

F

1.20

M

1.60

M

1.95

F

1.00

F

1.30

M

1.60

F

1.70

M

1.65

M

1.61

F

1.75

F

1.60

F

1.60

M

1.80

F

1.50

F

1.58

F

1.75

M

1.55

M

1.82

M

1.40

F

1.75

F

1.34

F

1.60

M

1.20

M

2.50

F

1.40

M

1.73

F

1.67

M

1.24

M

1.30

F

1.09

F

1.50

F

1.05

Data comparing year 7 and year 11 length estimates

How well can you estimate the length of a stick?

Introduction

I have collected data from year 7 and year 11 pupils and have recorded their estimates of how long they think the bamboo stick is. A bamboo stick of length 1.58m was held up in front of 173 year 7 pupils and 178 year 11 pupils. The stick was held horizontally so the length of the stick could not be compared to the height of the person. The pupils were then asked to estimate the length of the bamboo stick in metres. I have collated the data into two tables, one for year 7 and one for year 11. I have then used this data to pick a random sample of 50 from each year. I have given each set of measurements a number ranging from 1 to 173 for year 7 and 1 to 178 for year 11. In this way I will be able to use my calculator to obtain a random number, which I can use to select a set of data from my tables. I rounded the random number generated to the nearest integer. I have achieved this by using the following formula.

Year 7

Ran# x 173 = person on list.

Year 11

Ran# x 178 = person on list.

This has given me a sample of estimates to work with as shown on the following pages. I discarded the data if I got the same person on the list twice.

Year Eleven

Gender

Length (m)

F

1.27

M

1.50

F

1.55

M

2.60

M

1.24

M

1.20

F

1.40

M

1.75

F

1.45

M

1.35

M

1.56

M

1.48

F

1.44

F

1.70

M

1.30

M

1.63

F

1.60

M

1.50

F

1.60

F

1.75

M

1.77

M

1.35

M

1.65

M

1.42

F

1.60

M

1.45

M

2.00

F

1.58

F

1.25

M

1.30

M

1.45

M

1.75

M

1.48

F

1.45

F

1.45

F

0.75

F

1.10

F

1.90

F

1.66

F

1.50

M

1.55

F

1.63

M

1.49

M

1.40

M

1.50

M

1.65

M

1.32

F

1.80

F

1.52

M

1.70

Hypothesis

I believe that the year 11 pupils will have better estimates of the length of the stick and therefore the mean of their estimates will be closer to the actual length of the stick. I believe that the year 7 pupils will be less accurate and the mean of their estimates will be further away from the actual length of the stick. I also think that the year 11 data will have a smaller range with 50% guessing between 1.40m and 1.60m. I expect the year 11 data to deviate from the mean only slightly and therefore have a smaller standard deviation. I believe that the year 7 data will have a larger range with 10% guessing between 1.40m and 1.60m. I expect the year 7 data to have a larger standard deviation that the year 11 because I believe that each set of data will deviate from the mean by a large amount. I expect the bar chart of the year 11 data to look symmetrical and the median to be close to the actual length. I also expect the bar chart of the year 7 data to look symmetrical however I do not expect the mean to be as close to the actual length.

Overall I believe that the year 11 pupils will have more accurate estimates because they have been working with measurements for a longer time than the year 7 and therefore will have a better idea of how long a metre is.

Year 7

Estimated Length (m)

Frequency

Cumulative frequency

0 ≤ x < 0.2

0

0

0.2 ≤ x < 0.4

0

0

0.4 ≤ x < 0.6

0

0

0.6 ≤ x < 0.8

0

0

0.8 ≤ x < 1.0

0

0

1.0  ≤ x < 1.2

6

6

1.2  ≤ x < 1.4

10

16

1.4  ≤ x < 1.6

12

28

1.6  ≤ x < 1.8

14

42

1.8  ≤ x < 2.0

5

47

2.0  ≤ x < 2.2

1

48

2.2  ≤ x < 2.4

1

49

2.4  ≤ x < 2.6

1

50

2.6  ≤ x < 2.8

0

50

Year 11

Estimated Length (m)

Frequency

Cumulative frequency

0 ≤ x < 0.2

0

0

0.2 ≤ x < 0.4

0

0

0.4 ≤ x < 0.6

0

0

0.6 ≤ x < 0.8

1

1

0.8 ≤ x < 1.0

0

1

1.0  ≤ x < 1.2

1

2

1.2  ≤ x < 1.4

9

11

1.4  ≤ x < 1.6

21

32

1.6  ≤ x < 1.8

14

46

1.8  ≤ x < 2.0

2

48

2.0  ≤ x < 2.2

1

49

2.2  ≤ x < 2.4

0

49

2.4  ≤ x < 2.6

0

49

2.6  ≤ x < 2.8

1

50

Year 7

Estimated Length (m)

Frequency

Mid point

0 ≤ x < 0.2

0

0.1

0.2 ≤ x < 0.4

0

0.3

0.4 ≤ x < 0.6

0

0.5

0.6 ≤ x < 0.8

0

0.7

0.8 ≤ x < 1.0

0

0.9

1.0  ≤ x < 1.2

6

1.1

1.2  ≤ x < 1.4

10

1.3

1.4  ≤ x < 1.6

12

1.5

1.6  ≤ x < 1.8

14

1.7

1.8  ≤ x < 2.0

5

1.9

2.0  ≤ x < 2.2

1

2.1

2.2  ≤ x < 2.4

1

2.3

2.4  ≤ x < 2.6

1

2.5

2.6  ≤ x < 2.8

0

2.7

Year 11

Estimated Length (m)

Frequency

Mid point

0 ≤ x < 0.2

0

0.1

0.2 ≤ x < 0.4

0

0.3

0.4 ≤ x < 0.6

0

0.5

0.6 ≤ x < 0.8

1

0.7

0.8 ≤ x < 1.0

0

0.9

1.0  ≤ x < 1.2

1

1.1

1.2  ≤ x < 1.4

9

1.3

1.4  ≤ x < 1.6

21

1.5

1.6  ≤ x < 1.8

14

1.7

1.8  ≤ x < 2.0

2

1.9

2.0  ≤ x < 2.2

1

2.1

2.2  ≤ x < 2.4

0

2.3

2.4  ≤ x < 2.6

0

2.5

2.6  ≤ x < 2.8

1

2.7

Year 7 Cumulative Frequency

Median = ½ x total cumulative frequency

         = ½ x 50

         = 25th piece of data

Median = 1.56m

Upper Quartile = ¾ x total cumulative frequency

                 = ¾ x 50

                 =37.5th piece of data

Upper Quartile Value = 1.72m

Lower Quartile = ¼ x total cumulative frequency

                 = ¼ x 50

                 = 12.5th piece of data

Lower Quartile Value = 1.32m

Inter-quartile range = Upper Value – Lower Value

= 1.72 – 1.32

= 0.40m

Year 11 Cumulative Frequency

 Median = ½ x total cumulative frequency

         = ½ x 50

         = 25th piece of data

Median = 1.52m

Upper Quartile = ¾ x total cumulative frequency

                 = ¾ x 50

                 =37.5th piece of data

Upper Quartile Value = 1.68m

Lower Quartile = ¼ x total cumulative frequency

                 = ¼ x 50

                 = 12.5th piece of data

Lower Quartile Value = 1.40m

Inter-quartile range = Upper Value – Lower Value

= 1.68 – 1.40

= 0.28m

Gender

Length (m)

Deviation

Deviation Squared

F

1.27

-0.2558

0.06543364

M

1.50

-0.0258

0.00066564

F

1.55

0.0242

0.00058564

M

2.60

1.0742

1.15390564

M

1.24

-0.2858

0.08168164

M

1.20

-0.3258

0.10614564

F

1.40

-0.1258

0.01582564

M

1.75

0.2242

0.05026564

F

1.45

-0.0758

0.00574564

M

1.35

-0.1758

0.03090564

M

1.56

0.0342

0.00116964

M

1.48

-0.0458

0.00209764

F

1.44

-0.0858

0.00736164

F

1.70

0.1742

0.03034564

M

1.30

-0.2258

0.05098564

M

1.63

0.1042

0.01085764

F

1.60

0.0742

0.00550564

M

1.50

-0.0258

0.00066564

F

1.60

0.0742

0.00550564

F

1.75

0.2242

0.05026564

M

1.77

0.2442

0.05963364

M

1.35

-0.1758

0.03090564

M

1.65

0.1242

0.01542564

M

1.42

-0.1058

0.01119364

F

1.60

0.0742

0.00550564

M

1.45

-0.0758

0.00574564

M

2.00

0.4742

0.22486564

F

1.58

0.0542

0.00293764

F

1.25

-0.2758

0.07606564

M

1.30

-0.2258

0.05098564

M

1.45

...read more.

Middle

1.45

-0.0656

0.00430336

F

1.25

-0.2656

0.07054336

F

1.20

-0.3156

0.09960336

M

1.60

0.0844

0.00712336

M

1.95

0.4344

0.18870336

F

1.00

-0.5156

0.26584336

F

1.30

-0.2156

0.04648336

M

1.60

0.0844

0.00712336

F

1.70

0.1844

0.03400336

M

1.65

0.1344

0.01806336

M

1.61

0.0944

0.00891136

F

1.75

0.2344

0.05494336

F

1.60

0.0844

0.00712336

F

1.60

0.0844

0.00712336

M

1.80

0.2844

0.08088336

F

1.50

-0.0156

0.00024336

F

1.58

0.0644

0.00414736

F

1.75

0.2344

0.05494336

M

1.55

0.0344

0.00118336

M

1.82

0.3044

0.09265936

M

1.40

-0.1156

0.01336336

F

1.75

0.2344

0.05494336

F

1.34

-0.1756

0.03083536

F

1.60

0.0844

0.00712336

M

1.20

-0.3156

0.09960336

M

2.50

0.9844

0.96904336

F

1.40

-0.1156

0.01336336

M

1.73

0.2144

0.04596736

F

1.67

0.1544

0.02383936

M

1.24

-0.2756

0.07595536

M

1.30

-0.2156

0.04648336

F

1.09

-0.4256

0.18113536

F

1.50

-0.0156

0.00024336

F

1.05

-0.4656

0.21678336

Total =

75.78

4.77903200

Mean =

1.5156

Conclusion

After carrying out this investigation I have concluded that the year 11’s were more accurate in their estimates. This was because the mean is closer to the actual length and also because the standard deviation is smaller. This means that most people estimated close to the mean which links in with the fact that the inter-quartile range is only 0.28m. My estimate that 50% of year 11 would guess between 1.4 and 1.6m was a bit high as the actual percentage is 42%. I underestimated the year 7’s accuracy as I guessed that only 10% would estimate between 1.4 and 1.6m when in actual fact 24% of them did. The mean of both sets of data is very close with the year 7 mean being 1.52(to 1 d.p) and the year 11 mean being 1.53(to 1 d.p). This shows that both groups had similar guesses. However, the year 11 set of data had one person guessing a length of 0.75m and I think that this threw my data out, as it would have made the mean smaller. I have worked out the percentage errors for both years as shown below.

Year 7

Percentage error =  (mean – actual length)    x 100%image00.png

                            Actual length

                   =  (1.5156 – 1.58)       x 100%image01.png

                            1.58

                   =  -4% (to 1 s.f)

Year 11

Percentage error =  (mean – actual length)    x 100%image02.png

                            Actual length

                   =  (1.5258 – 1.58)       x 100%image01.png

                            1.58

Statistic

Year 7 Data

Year 11 Data

Mean

1.5156m

1.5258m

Median

1.56m

1.52m

Modal Group

1.6-1.8m

1.4-1.6m

Range

1.50m

1.85m

Inter-quartile range

0.40m

0.28m

Upper Quartile

1.72m

1.68m

Lower Quartile

1.32m

1.40m

Standard Deviation

0.309

0.259

Variance

0.095481

0.067081

Percentage error

-4%

-3%

                   =  -3% (to 1 s.f)

The percentage errors show that both the year 11 and year 7 mean was lower than the actual length of 1.58m. As you can see the difference between the means of the two sets of data is not significant however, overall when considering all the figures the year 11 pupils had better estimates.

Gender

Weight (Kg)

Deviation

Deviation Squared

F

0.375

-0.05556

0.0030869136

F

0.313

-0.11756

0.0138203536

M

1.500

1.06944

1.1437019136

M

0.300

-0.13056

0.0170459136

F

0.340

-0.09056

0.0082011136

F

0.150

-0.28056

0.0787139136

M

0.034

-0.39656

0.1572598336

M

0.765

0.33444

0.1118501136

M

0.667

0.23644

0.0559038736

M

0.274

-0.15656

0.0245110336

M

0.712

0.28144

0.0792084736

F

0.400

-0.03056

0.0009339136

M

0.600

0.16944

0.0287099136

M

0.300

-0.13056

0.0170459136

F

0.500

0.06944

0.0048219136

F

0.500

0.06944

0.0048219136

M

0.350

-0.08056

0.0064899136

M

1.130

0.69944

0.4892163136

F

0.571

0.14044

0.0197233936

F

0.598

0.16744

0.0280361536

M

0.432

0.00144

0.0000020736

M

0.372

-0.05856

0.0034292736

M

0.400

-0.03056

0.0009339136

M

0.360

-0.07056

0.0049787136

M

0.150

-0.28056

0.0787139136

F

0.210

-0.22056

0.0486467136

M

0.800

0.36944

0.1364859136

F

0.796

0.36544

0.1335463936

F

0.150

-0.28056

0.0787139136

M

0.350

-0.08056

0.0064899136

M

0.100

-0.33056

0.1092699136

F

0.020

-0.41056

0.1685595136

F

0.349

-0.08156

0.0066520336

F

0.500

0.06944

0.0048219136

F

0.250

-0.18056

0.0326019136

M

0.600

0.16944

0.0287099136

F

0.070

-0.36056

0.1300035136

M

0.100

-0.33056

0.1092699136

M

0.499

0.06844

0.0046840336

F

0.430

-0.00056

0.0000003136

F

0.570

0.13944

0.0194435136

M

0.380

-0.05056

0.0025563136

M

0.642

0.21144

0.0447068736

F

0.350

-0.08056

0.0064899136

M

0.600

0.16944

0.0287099136

M

0.380

-0.05056

0.0025563136

F

0.212

-0.21856

0.0477684736

F

0.225

-0.20556

0.0422549136

M

0.502

0.07144

0.0051036736

M

0.350

-0.08056

0.0064899136

Total =

21.528

3.5856963200

Mean =

0.43056

Year 11 Deviation

Year 7 Deviation

Gender

Weight (Kg)

Deviation

Deviation Squared

F

0.450

-0.08372

0.0070090384

M

0.777

0.24328

0.0591851584

F

0.200

-0.33372

0.1113690384

F

0.100

-0.43372

0.1881130384

F

0.465

-0.06872

0.0047224384

M

0.104

-0.42972

0.1846592784

F

0.707

0.17328

0.0300259584

M

0.200

-0.33372

0.1113690384

F

0.500

-0.03372

0.0011370384

F

0.856

0.32228

0.1038643984

F

0.460

-0.07372

0.0054346384

F

0.600

0.06628

0.0043930384

M

0.350

-0.18372

0.0337530384

F

0.612

0.07828

0.0061277584

F

0.800

0.26628

0.0709050384

F

0.900

0.36628

0.1341610384

M

0.750

0.21628

0.0467770384

F

0.500

-0.03372

0.0011370384

F

0.750

0.21628

0.0467770384

M

0.450

-0.08372

0.0070090384

M

0.475

-0.05872

0.0034480384

F

0.500

-0.03372

0.0011370384

F

0.400

-0.13372

0.0178810384

M

0.800

0.26628

0.0709050384

F

0.600

0.06628

0.0043930384

M

0.654

0.12028

0.0144672784

M

0.621

0.08728

0.0076177984

F

0.550

0.01628

0.0002650384

F

0.659

0.12528

0.0156950784

F

0.550

0.01628

0.0002650384

M

0.400

-0.13372

0.0178810384

F

0.325

-0.20872

0.0435640384

F

0.429

-0.10472

0.0109662784

F

0.400

-0.13372

0.0178810384

M

0.550

0.01628

0.0002650384

M

1.000

0.46628

0.2174170384

M

0.700

0.16628

0.0276490384

F

0.450

-0.08372

0.0070090384

F

0.480

-0.05372

0.0028858384

F

0.500

-0.03372

0.0011370384

M

0.350

-0.18372

0.0337530384

M

0.800

0.26628

0.0709050384

F

0.600

0.06628

0.0043930384

M

0.353

-0.18072

0.0326597184

F

0.652

0.11828

0.0139901584

M

0.167

-0.36672

0.1344835584

M

0.290

-0.24372

0.0593994384

F

0.500

-0.03372

0.0011370384

F

0.550

0.01628

0.0002650384

F

0.850

0.31628

0.1000330384

Total =

26.686

2.0916780800

Mean =

0.53372

...read more.

Conclusion

0

50

Year 11

Estimated Weight (Kg)

Frequency

Cumulative frequency

0.0 ≤ x < 0.1

3

3

0.1 ≤ x < 0.2

5

8

0.2 ≤ x < 0.3

5

13

0.3 ≤ x < 0.4

14

27

0.4 ≤ x < 0.5

5

32

0.5  ≤ x < 0.6

7

39

0.6  ≤ x < 0.7

5

44

0.7  ≤ x < 0.8

3

47

0.8  ≤ x < 0.9

1

48

0.9  ≤ x < 1.0

0

48

1.0  ≤ x < 1.1

0

48

1.1  ≤ x < 1.2

1

49

1.2  ≤ x < 1.3

0

49

1.3  ≤ x < 1.4

0

49

1.4  ≤ x < 1.5

0

49

1.5  ≤ x < 1.6

1

50

Year 7 Cumulative Frequency

Median = ½ x total cumulative frequency

         = ½ x 50

         = 25th piece of data

Median = 0.54Kg

Upper Quartile = ¾ x total cumulative frequency

                 = ¾ x 50

                 =37.5th piece of data

Upper Quartile Value = 0.69Kg

Lower Quartile = ¼ x total cumulative frequency

                 = ¼ x 50

                 = 12.5th piece of data

Lower Quartile Value = 0.43Kg

Inter-quartile range = Upper Value – Lower Value

= 0.69 – 0.43

= 0.26Kg

Year 11 Cumulative Frequency

 Median = ½ x total cumulative frequency

         = ½ x 50

         = 25th piece of data

Median = 0.38Kg

Upper Quartile = ¾ x total cumulative frequency

                 = ¾ x 50

                 =37.5th piece of data

Upper Quartile Value = 0.58Kg

Lower Quartile = ¼ x total cumulative frequency

                 = ¼ x 50

                 = 12.5th piece of data

Lower Quartile Value = 0.29Kg

Inter-quartile range = Upper Value – Lower Value

=0.58 – 0.29

= 0.29 Kg

Conclusion

Statistic

Year 7 Data

Year 11 Data

Mean

0.53372

0.43056

Median

0.54Kg

0.38Kg

Modal Group

0.4-0.5Kg

0.3-0.4Kg

Range

1.48Kg

0.9Kg

Inter-quartile range

0.26Kg

0.29Kg

Upper Quartile

0.69Kg

0.58Kg

Lower Quartile

0.43Kg

0.29Kg

Standard Deviation

0.205

0.268

Variance

0.42025

0.071824

Percentage Error

19%

-4%

After carrying out this extension to the investigation I have concluded that the year 11’s were more accurate in their estimates. The mean of the year 11 data is much closer to the actual weight than the mean of the year 7 data. This is because the year 11 data has a percentage error of only –4% whereas the year 7 data has a percentage error of 19%. However the year 11’s had a higher standard deviation. This shows that the year 7 greatly overestimated the weight while the year 11 slightly underestimated. The two scatter graphs comparing length and weight have very different patterns. The graph for the year 11 data shows a slight positive correlation while the graph for the year 7 data shows no correlation. This makes the year 7 estimates seem more random whereas the year 11 estimates are more logical. However both graphs are quite scattered with the year 11 graph being slightly more clustered than the graph of the year 7 data. When considering all the data shown in the table below I have found that the year 11s had more accurate estimates.

...read more.

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