• Join over 1.2 million students every month
  • Accelerate your learning by 29%
  • Unlimited access from just £6.99 per month

The normal distribution

Extracts from this document...

Introduction

The normal distribution

When many measures are taken of something (eg, scores in a test, people's heights, pollution levels in rivers) the spread of the values will have a bell shape, called the normal distribution.

A number of statistical tests use this characteristic distribution (or dispersion) of values to test whether two samples are the same or different.

There are several basic terms that are commonly used with the normal distribution.

Average (mean)

A measure of the average score in a set of data. The mean is found by adding up all the scores and then dividing by the number of scores.

Range

The difference between the largest core and the and smallest score.

Median

If a set of scores are arranged from lowest to highest the median is the score in the middle, with half above and half below.

Mode

The value that occurs most often

Standard deviation

s

A measure of the standard (average) deviation of the scores from the mean.The larger the standard deviation the larger the range of values/variation in the data

  1. Subtract each score from the mean
  2. Times each difference by itself (negs turn positive)
  3. Add up all the squared differences
  4. Divide the total by the number of scores minus 1
  5. Take the square root
...read more.

Middle

image07.png

image08.png

Then calculate the square root to get the standard deviation

Comparing two samples: using the t test

The average, standard deviation and the number of scores in each sample are the three things needed to do a test. A t test is used with two samples of data to test whether they are significantly different (ie, whether one is truly higher or lower than the other). The same sample of scores as used above is now compared with another sample of scores.

Sample 1 scores

Sample 2 scores

41

38

43

32

37.5

35.5

38.5

33

44

31.5

38

40.5

37.5

34

Average (image09.png)

39.93

34.93

Standard deviation (s)

2.73

3.31

Number of scores (n)

7

7

image11.pngimage01.pngimage02.pngimage10.png

  1. Put the values into the equation and work it out carefully!
  2. Note down the value of t found. In this case it is 3.08.
  3. You will also need to know how many degrees of freedom to use with the critical values of t table. Degrees of freedom  =  (nsample1 + nsample2) – 2 . In this example  this equals 7 + 7 –2 = 12.
  4. Find the value of t
...read more.

Conclusion

png" style="width:648px;height:469.47px;margin-left:0px;margin-top:0px;" alt="image03.png" />

2. Calculate the average, range, median and mode for the following set of data (a random set of your exam results from the last exam): 66.25, 15, 32.5, 26.25, 48.75, 48.75, 36.25, 35, 68.75, 72.5, 43.75, 40, 20, 48.75, 12.5, 41.25, 53.75, 50, 31.25, 95, 22.5, 33.75, 27.5, 55, 12.5, 45, 18.75, 42.5, 62.5, 85, 75

Degrees of freedom

Value of t that must be exceeded (5% level)

1

12.706

2

4.303

3

3.182

4

2.776

5

2.571

6

2.447

7

2.365

8

2.306

9

2.262

10

2.228

11

2.201

12

2.179

13

2.160

14

2.145

15

2.131

16

2.120

17

2.110

18

2.101

19

2.093

20

2.086

22

2.074

24

2.064

26

2.056

28

2.048

30

2.042

40

2.021

60

2.000

120

1.980

3. The two sets of data given below are resting heart rates for a group of students and a group of professional athletes. Use the t test to find out if they are significantly different (using the table at right to test the value of t with the appropriate number of degrees of freedom).  I need to see how the mean, standard deviation and t value were calculated.

        Professional

Students                athletes

57.1                        61.7

47.6                        47.0

58.0                        55.5

74.8                        62.6

  1. 41.8

51.9                        60.8

64.2                        50.2

49.6                        44.2

67.2                        45.4

62.6                        39.3

...read more.

This student written piece of work is one of many that can be found in our AS and A Level Probability & Statistics section.

Found what you're looking for?

  • Start learning 29% faster today
  • 150,000+ documents available
  • Just £6.99 a month

Not the one? Search for your essay title...
  • Join over 1.2 million students every month
  • Accelerate your learning by 29%
  • Unlimited access from just £6.99 per month

See related essaysSee related essays

Related AS and A Level Probability & Statistics essays

  1. The mathematical genii apply their Statistical Wizardry to Basketball

    I can say now that skill level did not increase during the collection of the sample size but what is more likely to have occurred is the opposite. The explanation for Dom being more tired, bored or frustrated is probably because he shot a total of 345 baskets whereas Lee completed his in 269 shots.

  2. Driving test

    The results show that with my formula drivers 63, 74 and 216 failed their test. Only driver 118 passed. Hypothesis 1 ("The more lessons taken by a pupil the fewer mistakes they make in the test') is correct. This means that my prediction also correct.

  1. Estimating the length of a line and the size of an angle.

    Quarter 1st quarter 2nd quarter 3rd quarter 4th quarter Students 62, 14, 34 1, 33, 74 17, 13, 5 24, 73 Random number results from year 11 males Quarter 1st quarter 2nd quarter 3rd quarter 4th quarter Students 194, 102, 121, 83, 162 178, 96, 92, 181 94, 85, 111,

  2. "The lengths of lines are easier to guess than angles. Also, that year 11's ...

    as their modal class was the correct class where the actual angle length could be found. Line Year 9: <6cm class interval Year 11: <6cm class interval again. These show that the majority of people in both year groups guessed around the same number, but it was not the correct class interval of <5cm.

  1. find out if there is a connection between people's IQ and their average KS2 ...

    All I had to do was select the two columns from the data that I wanted - IQ and average KS2 SATs results and the graph was drawn. As you can see from my scatter graph, nearly all the points are bunched up, very near the trend line on the lower middle part of the graph.

  2. I want to find out if there is a connection between people's IQ and ...

    All I had to do was select the two columns from the data that I wanted - IQ and average KS2 SATs results and the graph was drawn. As you can see from my scatter graph, nearly all the points are bunched up, very near the trend line on the lower middle part of the graph.

  1. Investigation into the relationship between P1 exam results and A-level results

    6 94 90 7 62 64 8 90 73 9 87 84 10 84 82 11 76 74 12 62 78 13 92 70 14 90 88 15 92 84 16 73 63 17 48 55 18 69 69 19 76 70 20 100 86 21 92 80 22 58

  2. Is there a Correlation between GCSE Mathematics and English Literature scores?

    The population for my investigation is the 1998 Year 11 class from Thistley Hough High School which comprised of 162 pupils including myself. As GCSE grades are non numerical data I simply used the GCSE points scored, i.e.-A*=8, A=7, B=6.....U=0.

  • Over 160,000 pieces
    of student written work
  • Annotated by
    experienced teachers
  • Ideas and feedback to
    improve your own work