The Normal Distribution Name:________________________

What is the normal distribution? It is a bell shaped curve which is used to characterise large amounts of statistical information. For example, when the heights or weight of all male people in Ireland are considered. A symmetrical pattern will form where one data piece of data shows up more than others.

This diagram is known as a symmetrical or normal distribution. Within this the mean, the mode and the median will all have the same value.

Normal distribution terminology In the normal distribution, the mean is always the central value and is represented by the sign The standard deviation tells us how much variation or change there is from the mean. The sign for the standard deviation is

Normal distribution terminology This is no different for the normal distribution. If we move to the left or to the right of the mean then we are said to be moving a ‘certain number of standard deviations away from the mean’ E.g if I move 2 steps to the right then I move 2 standard deviations to the right of the mean.

The Normal Distribution and the Empirical rule

The Empirical Rule The empirical rule sets out 3 simple rules for when we move a certain number of standard deviations away from the mean. When using the empirical rule it is good to refer back to height and weight.

Empirical Rule Part 1 About 68% of the data lies within 1 standard deviation of the mean. Formula =

Empirical Rule Part 2 About 95% of the data lies within 2 standard deviations of the mean. Formula=

Empirical Rule Part 3 About 99.7% of the data lies within 3 standard deviations of the mean. Formula

Question A normal distribution has a mean of 60 and a standard deviation of 5. 1.Find the range within 68% of the data lies? 2.Find the range within 95% of the data lies? 3.Find the range within 99.7% of the data lies?

Question The heights of a large sample of adults are normally distributed with a mean of 170cm and a standard deviation of 8cm. Within what do limits does 68% of the heights lie? Within what do limits does 99.7% of the heights lie?

Question The marks out of a 100 in an IQ test are normally distributed. If the mean mark is 60 and the standard deviation is 6 marks. What percentage of marks lie between 48 and 72 marks? What percentage of the marks lie between 42 and 78 marks?

The Standard Normal distribution Curve What is the mean and standard deviation of the standard normal distribution? How much of the data lies within 1 standard deviation of the mean? How much of the data lies within 2 standard deviation of the mean? How much of the data lies within 3 standard deviation of the mean? What is P(z = 3) 68% 96% 99.7% 0 0

The heights of women is Normally distributed with a mean of 168cm and a standard deviation of 7cm. Between what heights do (a) 68% of women (b) 96% of women (c) 99.7% of women lie? (a)68% of women lie between 161cm and 175cm (b) 96% of women lie between 154cm and 182cm (c) 99.7% of women lie between 147cm and 189cm (a)68% of women lie between 161cm and 175cm (b) 96% of women lie between 154cm and 182cm (c) 99.7% of women lie between 147cm and 189cm If you had a large number of results that were outside these bounds it could mean that -The sample size was too small -The assumption that it is normally distributed was wrong! If you had a large number of results that were outside these bounds it could mean that -The sample size was too small -The assumption that it is normally distributed was wrong!

Properties of a normal distribution curve 1.We can use the normal distribution to calculate probabilities. 2.The curve is symmetric at the mean which tells us that 50% of the data lies on the left of the mean and 50% lies on the right of the mean. 3.The area under the curve is equal to 1. 4.The mean, mode and median of this curve are all the same

Z Scores z – scores tell us the position of a score in relation to the mean using the standard deviation as a unit of measurement. The z – score is the number of standard deviations by which the score departs from the mean. E.g a Z Score of 1 means one standard deviation from the mean.

Questions which involve standard normal distributions. Key Point: We can only use our standard normal curve for z scores between -3 and 3.

Steps in completing normal distributions questions. 1.Ensure the Z Score is between -3 and 3. 2.If it is not between -3 and 3 then we will have standardise this value. 3.Draw a sketch of the bell curve and shade in the area you are looking for. Remember we are always looking for the area to the left of our z score. Remember the area under the curve is always 1 in a standard normal curve. 4. Refer to your normal distribution tables to find the probability

How to read the Normal distribution table P(z) means the area under the curve on the left of z

How to read the Normal distribution table Z= 0.24 means the area under the curve on the left of 0.24 and is this value here:

Now find the following use your tables to find the following probabilities:

Values of P(z) P(z > 0.8)=

Now find the following use your tables to find the following probabilities:

P(z < -1.5) =1- P(Z < 1.5)

Now find the following use your tables to find the following probabilities:

P(-1< z < 1.5)

Now find the following use your tables to find the following probabilities:

finding probabilities By drawing a picture and using the Normal distribution tables calculate the following: (a)P(z ≤ 1.23) (b)P(z ≥ 2.47) (c)P(z ≤ -1.2) (d)P(z ≥ -0.47) (e)P(1.23 ≤ z ≤ 2.45)

Solutions By drawing a picture and using the Normal distribution table on page 24 calculate the following: (a)P(z ≤ 1.23) = (b)P(z ≥ 2.47) = 1 – P(z ≤ 2.47) = 1 – = (a)P(z ≤ -1.2) = P(z ≥ 1.2) = 1 – P(z ≤ 1.2) = 1 – = (a)P(z ≥ -0.47) = P(z ≤ 0.47) = (a)P(1.23 ≤ z ≤ 2.45) = P(z ≤ 2.45) – P(z ≤ 1.23) = – =

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