1. SADC Course in Statistics The normal distribution (Session 08)

2. To put your footer here go to View > Header and Footer 2 Learning Objectives At the end of this session you will be able to: describe the normal probability distribution state and interpret parameters associated with the normal distribution use a calculator and statistical tables to calculate normal probabilities appreciate the value of the normal distribution in practical situations

3. To put your footer here go to View > Header and Footer 3 The Normal Distribution In the previous two sessions, you were introduced to two discrete distributions, the Binomial and the Poisson. In this session, we introduce the Normal Distribution – one of the commonest distributions followed by a continuous random variable For example, heights of persons, their blood pressure, time taken for banana plants to grow, weights of animals, are likely to follow a normal distribution

4. To put your footer here go to View > Header and Footer 4 Example: Weights of maize cobs Graph shows histogram of 100 maize cobs. Data which follows the bell shape of this histogram are said to follow a normal distribution.

5. To put your footer here go to View > Header and Footer 5 Frequency definition of probability In a histogram, the bar areas correspond to frequencies. For example, there are 3 maize cobs with weight < 100 gms, and 19 maize cobs with weight < 120 gms. Hence, using the frequency approach to probability, we can say that Prob(X<120) = 19/100 = 0.019 The areas under the curve can be regarded as representing probabilities since the curve and edges of histogram would coincide for n= .

6. To put your footer here go to View > Header and Footer 6 Probability Distribution Function The mathematical expression describing the form of the normal distribution is f(x) = exp(–(x–) 2 /2 2 )/(2 2 ) Two parameters associated with the normal distribution, its mean  and variance  2.

7. To put your footer here go to View > Header and Footer 7 Properties of the Normal Distribution Total area under the curve is 1 characterised by mean & variance: N(, 2 ) symmetric about mean () 95% of observations lie within ± 2 of mean

8. To put your footer here go to View > Header and Footer 8 The Standard Normal Distribution This is a distribution with =0 and =1, shown below in comparison with N(0, 2 ), =3.

9. To put your footer here go to View > Header and Footer 9 The Standard Normal Distribution This is a distribution with =0 and =1. Tables give probabilities associated with this distribution, i.e. for every value of a random variable Z which has a standard normal distribution, values of Pr(Z<z) are tabulated. In graph on right, P=Pr(Z<z). Symmetry means any area (prob) can be found.

10. To put your footer here go to View > Header and Footer 10 Calculating normal probabilities Any random variable, say X, having a normal distribution with mean  and standard deviation , can be converted to a value (say z) from the standard normal distribution. This is done using the formula z = (X - ) /  The z values are called z-scores. The z scores can be used to compute probabilities associated with X.

11. To put your footer here go to View > Header and Footer 11 An example The pulse rate (say X) of healthy individuals is expected to have a normal distribution with mean of 75 beats per minute and a standard deviation of 8. What is the chance that a randomly selected individual will have a pulse rate < 65? We need to find Pr(X < 65) i.e. Pr(X - 75 < 65 - 75) = Pr[ (X – 75/8) < (-10/8) ] = Pr(Z < -1.28)  Pr(Z<-1.3) = 0.0968 (using tables of the standard normal dist n )

12. To put your footer here go to View > Header and Footer 12 A practical application Malnutrition amongst children is generally measured by comparing their weight-for-age with that of a standard, age-specific reference distribution for well-nourished children. A child’s weight-for-age is converted to a standardised normal score (an z-score), standardised to  and  of the reference distribution for the child’s gender and age. Children whose z-score<-2 are regarded as being underweight.

13. To put your footer here go to View > Header and Footer 13 A Class Exercise Similarly to the above, height-for-age is used as a measure of stunting again converted to a standardised z score (stunted if z-score<-2). Suppose for example, the reference distribution for 32 months old girls has mean 91 cms with standard deviation 3.6 cms. What is the probability that a randomly selected girl of 32 months will have height between 83.8 and 87.4 cms? Graph below shows the area required. A class discussion will follow to get the answer.

14. To put your footer here go to View > Header and Footer 14 Depicting required probability as an area under the normal curve Answer =

15. To put your footer here go to View > Header and Footer 15 Is a child stunted? Suppose a 32 month old girl has height-for- age value = 82.1 Would you consider this child to be stunted? Discuss this question with your neighbour and write down your answer below.

16. To put your footer here go to View > Header and Footer 16 Cumulative normal distribution In example above, the shaded area is 0.6, the value of a from tables of the standard normal distribution is 0.726. Cumulative distribution is given by the function F(x) = P(X ≤a)

17. To put your footer here go to View > Header and Footer 17 P(a<X<b) = F(b)-F(a) is the area under the cumulative normal curve between points a and b.

18. To put your footer here go to View > Header and Footer 18 Practical work follows …