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15. Normal Distribution

Check out the overview section for relevant importance of this section compared to other topics in this course. In 2020 exam, two questions appeared from this topic making up 8 marks (2020Q82020Q35). This section covers the following parts of the syllabus:

The infographic below shows all the past exam questions from 2010 to 2020 relevant to this topic sorted by difficulty level and further broken down into sub-topics. This will form the foundation of our study as we would like you to focus first on the easy questions and quickly develop skills to get those easy marks and then challenge yourself with the harder ones. z-score:

Before talking about z-score, it is good to first start with a bit of refresher of a few things you are expected to know:

• Mean: This is the average value, so the mean of values 6, 7, 4, 7, 8 is (6 + 7 + 4 + 7 + 8)/5 = 6.4.
• Standard Deviation: This value indicates deviations or how spread out values are around the mean. For example, another sample with values 2, 26, 1, 2, 1 also has the mean of 6.4 but the sample values are more far apart from the mean; hence a higher standard deviation.
• Median: This is the middle value of the range, so in the first sample above, if you line up values in ascending order 4, 6, 7, 7, 8, then the middle value is the 3rd observation which is 7.
• Mode: This is the value that occurs the most which is 7 in this case.

Now coming to z-score. This is also called the standardised score.

What is the need for this z-score?

Many times, observation values are not comparable when taken from different samples. For example, a student scored 65 marks in an exam. Knowing this in itself is not sufficient to determine whether it is a good or bad mark.

What if the average score of all students was 70, then it is probably not such a great mark, but if the average score of the other students is 50, then it is a good mark.

Only comparing with the average mark is not enough. Consider the following two scenarios where the average mark is 70 and Student 1 has 65 marks: In both scenarios, Student 1’s mark is 65 which is below the average mark of 70. However, in the first scenario, the relative position of Student 1 is not as bad as in the second scenario.

Only comparing with the average mark in the above situation will not capture this difference. A way of capturing this is through the standardised score or z-score where the difference from the mean is divided by the standard deviation to give a standard value for comparison.

So, z-score for scenario 1 will be -0.283, while in scenario 2 it will be -1.414, indicating in scenario 2 the student is further away from the average than the others in the group. Example 1

Alan has the following scores:

• 45 in Mathematics where the mean score of class is 43 and the standard deviation of 5
• z-score of 0.667 in English where the mean score of class is 27 and the standard deviation of 12
• 37 in Physics with a standardised score of 0.45 where the standard deviation of the class is 9
1. What is Alan’s z-score in Mathematics?
2. What is Alan’s actual score in English?
3. What is the mean score of the class in Physics?
4. Which course did he do best?

Normal Distribution:

A bit of basics again before talking about Normal Distribution. A frequency graph is a representation of frequency distribution where on one axis has observation values and other axis has the associated probability.

Normal distribution is a special kind of frequency distribution that has some unique features.   Example 2

Which one of the following distributions has the highest mean and the highest standard deviation? Empirical Rule:

If you are given a normal distribution, there are a few things you can automatically assume, and this is what empirical rules are. If ever you are stuck in a question, always try to draw out a normal distribution curve and work out the area question is asking about.

Probability using table:

You may sometimes be given a probability table and you have to use it to determine the probability.

The table will be most likely for standard normal distribution (Mean = 0 and Std. Dev. = 1) and its values would represent the area on the left side of z-score. Make sure to read the title of the table to confirm.

Solving these questions is simply about determining the z-score and then looking up values from the table and determining the required probability.

Get access to 20 Mock Exams with over 700 exam-style questions for HSC Standard Maths. 