C82MCP Diploma Statistics School of Psychology University of Nottingham 1 Overview Central Limit Theorem The Normal Distribution The Standardised Normal Distribution Z Scores Estimation Confidence Intervals

C82MCP Diploma Statistics School of Psychology University of Nottingham 2 Central Limit Theorem The sampling distribution of the means of samples becomes normal as the sample size increases. The sampling distribution of the mean for sufficiently large samples will be normal (N>30). The sampling distribution of the mean will always be normal if the underlying population is also normal irrespective of sample sizes. For small samples (N< 30) taken from a normally distribution population, we know the form of the sampling distribution of the mean.

C82MCP Diploma Statistics School of Psychology University of Nottingham 3 The Normal Distribution The normal distribution is a specific distribution with a particular shape It is mathematically defined by the expression This defines the probability x with respect to two parameters, the mean and the variance of a population of scores

C82MCP Diploma Statistics School of Psychology University of Nottingham 4 Standardised Normal Distribution We can define a normal distribution of standardised scores in the following way: where z is known as a standardised score

C82MCP Diploma Statistics School of Psychology University of Nottingham 5 Standardised Normal Distribution Using calculus (specifically integration) we can calculate the area under a standardised normal distribution This tells us the proportion of scores that fall under a particular area of the distribution. We can use tables of standardised scores to estimate the probability of a particular score occurring in any set of data

C82MCP Diploma Statistics School of Psychology University of Nottingham 6 Using Z Scores To use the standardised normal distribution we must adopt the basic assumption that the population of scores is normally distributed What proportion of IQ scores are greater than 125? IQ scores are normally distributed with a mean of 100 and a standard deviation of 15

C82MCP Diploma Statistics School of Psychology University of Nottingham 7 Using Z Scores We have to calculate the z score associated with an IQ of 125. To do this we calculate the difference between the mean and the score and divide by the standard deviation We obtain:

C82MCP Diploma Statistics School of Psychology University of Nottingham 8 Using Z Scores Now we look at the tables to find what proportion of scores are beyond the z score of Looking at the table entry for z = 1.67 we find that of the area of the curve lies beyond the z value of If we multiply this by 100 we obtain the percentage of scores that lie beyond this value. In other words 100 x = 4.745% of the population have an IQ of greater than 125

C82MCP Diploma Statistics School of Psychology University of Nottingham 9 Using Z Scores What proportion of IQ scores are less than 60? We have to calculate the z score associated with an IQ of 60. First calculate the difference between the mean and the score and divide by the standard deviation We obtain:

C82MCP Diploma Statistics School of Psychology University of Nottingham 10 Using Z Scores Now we look at the tables to find what proportion of scores are beyond the z score of Looking at the table entry for z = 2.67 we find that of the area of the curve lies beyond the z value of If we multiply this by 100 we obtain the percentage of scores that lie beyond this value. In other words 100 x = 0.378% of the population have an IQ of less than 60

C82MCP Diploma Statistics School of Psychology University of Nottingham 11 Using Z Scores What proportion of scores lie between 85 and 115 on the IQ scale To do this we calculate the difference between the mean and the score and divide by the standard deviation for both points We get:

C82MCP Diploma Statistics School of Psychology University of Nottingham 12 Using Z Scores Now we look at the tables to find what proportion of scores are between the mean of the distribution and and 1.00 respectively. Looking at the table entry for z = we find that of the area of the curve lies between the mean the z score Looking at the table entry for z = 1.00 we find that of the area of the curve lies between the mean the z score The total proportion is the addition of the two values, i.e = In other words 100 x = % of the population have an IQ of between 85 and 115.

C82MCP Diploma Statistics School of Psychology University of Nottingham 13 Estimation Most of the time we do not know about population parameters We would like to be able to make a judgement about the population parameters In parametric statistics we can make "best guess" judgements about the parameters of populations These "best guesses" are known are estimates

C82MCP Diploma Statistics School of Psychology University of Nottingham 14 Point & Interval Estimation There are two kinds of estimates, point and interval. With point estimates we attempt to assign a particular value to a population parameter such as the mean or the variance With interval estimates we try and construct a range in which the population parameter might fall and to which we can attach a probability

C82MCP Diploma Statistics School of Psychology University of Nottingham 15 Point Estimates For an estimate to be considered a good estimate then it must be unbiased, sufficient and consistent. Unbiased The mean of the sampling distribution is equal to the population parameter being estimated. Sufficient The statistic on which the estimate is based uses all the information in the sample. Consistent Based on a statistics whose accuracy increases as sample size increases.

C82MCP Diploma Statistics School of Psychology University of Nottingham 16 Measures of Centre All measures of centre i.e. mean, mode and median are unbiased measures of their respective parameters. The mean, however, is the only one of the sample statistics which is both sufficient and consistent.

C82MCP Diploma Statistics School of Psychology University of Nottingham 17 Measures of Spread Both the variance and the standard deviation are biased estimates of the population parameters  and  2 The mean of the sampling distribution of the variance is too small as an estimate of the population variance by a factor of: so that is an unbiased estimate of  2

C82MCP Diploma Statistics School of Psychology University of Nottingham 18 Measures of spread Too distinguish the sample variance and the sample based estimate of the population variance we will refer to the sample based estimate as: Similarly the sample based estimate of the population standard deviation is referred to as:

C82MCP Diploma Statistics School of Psychology University of Nottingham 19 Standard Error of the Mean The population standard error of the mean is defined as: The sample based estimate of the population standard error of the mean is defined as:

C82MCP Diploma Statistics School of Psychology University of Nottingham 20 Interval Estimates Interval estimates are calculated on the basis of three factors: A point estimate for the parameter A measure of spread in the population A probability value

C82MCP Diploma Statistics School of Psychology University of Nottingham 21 Confidence Intervals Suppose that we have tested the IQ of a number of subjects in an experiment. We are going to calculate the 95% confidence interval for the population mean, µ First we have to compute the population standard error of the mean: The standard error of the mean of the population tells us how much we can expect the population mean to fluctuate.

C82MCP Diploma Statistics School of Psychology University of Nottingham 22 Confidence Intervals Assuming the population is normally distributed means that the sampling distribution of the mean is also normally distributed (central limits theorem). Now define an area of the sampling distribution that should contain the middle 95% of the possible sample means. In order to do this we must use the standardised formula as applied to the sampling distribution:

C82MCP Diploma Statistics School of Psychology University of Nottingham 23 Confidence Intervals If we look at the tables of z values we can find that the centre based z scores that include 95% of the distribution is equal to ±1.96. The upper limit of our range of values for the population mean is: The lower limit of our range of values for the population mean is:

C82MCP Diploma Statistics School of Psychology University of Nottingham 24 Confidence Intervals In general terms, the formula for the 95% interval for µ is: For any level of confidence, ranging from 0 to % we have: