DS5 The Normal Distribution Write down all you can remember about the mean, median, mode, and standard deviation. 1
DS5 The Normal Distribution Basic concepts: Describing and using Z scores calculated using the formula Comparing data from different sets of scores Understanding the relevance of Z scores and percentages between Z scores 2
DS5 The Normal Distribution Just say that you have been studying very hard in General Maths, and Mrs Woodley says you are improving...on your half yearly exam you got 88% but in your trial you only got 63%. You begin to think you may not have done very well after all...or was it that the trial exam was too hard? You compare with 11 of your classmates with both sets of marks: Half Yearly Trial 95, 85, 80, 84, 78, 80 50, 60, 62, 63, 55, 56 82, 92, 82, 88, 77, 85 60, 55, 60, 65, 60, 62 We can compare your result with the mean to find a measure of the difficulty level of the test. Exam Mean Your mark Comment rub and reveal In both test, your mark was 4 marks above the mean, so you did fairly well. Consider the standard deviations by looking at the dot plots of both tests: your mark your mark exam S.D. your mark above the mean comment As your mark was <1 S.D. above the mean in the half yearly and >1 S.D. above the mean in the trial, your trial mark is BETTER! pull 3
DA6 DS5 The Normal Distribution Results can be compared using a combination of the mean and the standard deviation. We use Z scores to do this; a Z score simply means how many standard deviations a score is above the mean. A Z score of 2 means that the score is exactly 2 standard deviations above the mean. A Z score of 1.5 means the score is 1.5 standard deviations below the mean. How are Z scores calculated? Z scores fade... Don't forget that there are two types of standard deviation...σ n for the population standard deviation and σ n 1 for the sample standard deviation; but we use only the population standard deviation. 4
DA6 DS5 The Normal Distribution e.g. In an English exam, the mean mark was 55 and the standard deviation was 6. What Z score corresponds to a mark of: a) 61? b) 43? c) 51? You are given the standard deviation (no need to work it out). If it's an easy question (like a and b are here), you may be able to answer by observation. You can always use the formula, though. a) 61 is exactly 6 marks above the mean ( 1 S.D.) so the Z score is 1 b) 43 is exactly 12 marks below the mean (2 S.D.) so the Z score is 2 c) For this question, use the formula: Z = x x s = 51 55 6 = 0.67 so the Z score is 0.67 5
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DA6 DS5 The Normal Distribution How can Z scores be used? Just say you scored 62 in Biology, 64 in English and 68 in Music. How you actually went depends on the mean and standard deviation for each exam. The statistics for each exam are given below: By either using the formula or by observation, it can be seen that your Biology mark has a Z score of 0.5, English has a Z score of 2, and Music a Z score of 1. Put the subjects in order from where you did the best, to where you need to study more! fade... English Biology Music 9
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DA6 DS5 The Normal Distribution When you perform a large number of trials (experiments), your results will over time approach what is called a 'Normal Distribution'. This looks like a bell shaped curve, as in the diagram below. online activity simulation x 3σ x 2σ x σ x x + σ x + 2σ x + 3σ online activity simulation 2 The main pieces of information to take from this diagram are: 1. 68% of scores lie within 1 σ of the mean (have a Z score between 1 and 1) 2. 95% of scores lie within 2 σ of the mean (have a Z score between 2 and 2) 3. 99.7% of scores lie within 3 σ of the mean (have a Z score between 3 and 3) Note: scores lying outside 3 σ from the mean are referred to as 'outliers' 21
DA6 DS5 The Normal Distribution Experiment: Roll two normal dice 40 times and record the total each time (use a table). Draw a column graph of your results. We will combine all our samples to get a full class distribution. 22
DA6 DS5 The Normal Distribution Paper clips come in boxes of 100 with a standard deviation of 5 clips per box. Assuming that the population has a normal distribution: a) What percentage of boxes have between 95 and 105 clips? b) How many stickers will be in 95% of boxes? a) We know the mean is 100 and the standard deviation is 5. Think back to the normal distribution curve 95 is 1 σ below the mean and 105 is 1 σ above the mean. This means that 68% of all scores lie between 95 and 105. b) 95% of all boxes will contain between 90 and 110 clips, as these figures are 2 σ from the mean (on either side). pull pull 23
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