Discovering Statistics Using IBM SPSS Statistics: North American Edition

by Andy Field

A population can be very general (all human beings) or very narrow (all male ginger cats called Bob).

we rarely, if ever, have access to every member of a population

Therefore, we collect data from a smaller subset of the population known as a sample (the scaled-down bridge) and use these data to infer things about the population as a whole.

The bigger the sample, the more likely it is to reflect the whole population.

Remember that parameters are the ‘P’ in the SPINE of statistics.

Statistical models are made up of variables and parameters.

parameters are not measured and are (usually) constants believed to represent some fundamental truth about the relations between variables in the model.

we can predict values of an outcome variable based on a model. The form of the model changes, but there will always be some error in prediction, and there will always be parameters that tell us about the shape or form of the model.

When you see equations where these little hats are used, try not to be confused, all the hats are doing is making explicit that the values underneath them are estimates.

It’s important to assess the fit of any statistical model

With most statistical models we can determine whether the model represents the data well by looking at how different the scores we observed in the data are from the values that the model predicts.

the error or deviance for a particular entity is the score predicted by the model for that entity subtracted from the corresponding observed score.

The line representing the mean can be thought of as our model, and the dots are the observed data. The diagram also has a series of vertical lines that connect each observed value to the mean value. These lines represent the error or deviance of the model

although the sum of squared errors (SS) is a good measure of the accuracy of our model, it depends upon the quantity of data that has been collected – the more data points, the higher the SS.

To compute the average error we divide the sum of squares (i.e., the total error) by the number of values (N) that we used to compute that total.

To estimate the mean error in the population we need to divide not by the number of scores contributing to the total, but by the degrees of freedom (df), which is the number of scores used to compute the total adjusted for the fact that we’re trying to estimate the population value

we can use the sum of squared errors and the mean squared error to assess the fit of a model. The mean squared error is also known as the variance. As such, the variance is a special case of a more general principle that we can apply to more complex models, which is that the fit of the model can be assessed with either the sum of squared errors or the mean squared error. Both measures give us an idea of how well a model fits the data: large values relative to the model indicate a lack of fit.

We have seen that models are defined by parameters, and these parameters need to be estimated from the data that we collect.

Estimation is the ‘E’ in the SPINE of statistics. We used an example of the mean because it was familiar, but it will also illustrate a general principle about how parameters are estimated.

Throughout this book, we will fit lots of different models, with parameters other than the mean that need to be estimated. Although the equations for estimating these parameters will differ from that of the mean, they are based on this principle of minimizing error: they will give you the parameter that has the least error given the data you have.

We have looked at how we can fit a statistical model to a set of observations to summarize those data. It’s one thing to summarize the data that you have actually collected, but in Chapter 1 we saw that good theories should say something about the wider world.

we need to go beyond the data, and to go beyond the data we need to begin to look at how representative our samples are of the population of interest. This idea brings us to the ‘S’ in the SPINE of statistics: the standard error.

the standard deviation tells us about how well the mean represents the sample data. However, if we’re using the sample mean to estimate this parameter in the population, then we need to know how well it represents the value in the population, especially because samples from a population differ.

the sample mean is different in the second sample than in the first. This difference illustrates sampling variation: that is, samples vary because they contain different members of the population;

If we plotted the resulting sample means as a frequency distribution or histogram,7 we would see that three samples had a mean of 3, means of 2 and 4 occurred in two samples each, and means of 1 and 5 occurred in only one sample each. The end result is a nice symmetrical distribution known as a sampling distribution. A sampling distribution is the frequency distribution of sample means (or whatever parameter you’re trying to estimate) from the same population.

The sampling distribution of the mean tells us about the behavior of samples from the population, and you’ll notice that it is centerd at the same value as the mean of the population (i.e., 3). Therefore, if we took the average of all sample means we’d get the value of the population mean. We can use the sampling distribution to tell us how representative a sample is of the population.

If our ‘observed data’ are sample means then the standard deviation of these sample means would similarly tell us how widely spread (i.e., how representative) sample means are around their average. Bearing in mind that the average of the sample means is the same as the population mean, the standard deviation of the sample means would therefore tell us how widely sample means are spread around the population mean: put another way, it tells us whether sample means are typically representative of the population mean.

The standard deviation of sample means is known as the standard error of the mean (SE) or standard error for short.

the square root of this value would need to be taken to get the standard deviation of sample means: the standard error.

Some exceptionally clever statisticians have demonstrated something called the central limit theorem, which tells us that as samples get large (usually defined as greater than 30), the sampling distribution has a normal distribution with a mean equal to the population mean, and a standard deviation shown in equation (2.14):

When the sample is relatively small (fewer than 30) the sampling distribution is not normal: it has a different shape, known as a t-distribution,

The ‘I’ in the SPINE of statistics is for ‘interval’; confidence interval, to be precise.

We can also use this information to calculate boundaries within which we believe the population value will fall. Such boundaries are called confidence intervals.

we could use an interval estimate instead: we use our sample value as the midpoint, but set a lower and upper limit as well.

The crucial thing is to construct the intervals in such a way that they tell us something useful.

Typically, we look at 95% confidence intervals, and sometimes 99% confidence intervals, but they all have a similar interpretation: they are limits constructed such that, for a certain percentage of samples

To calculate the confidence interval, we need to know the limits within which 95% of sample means will fall. We know (in large samples) that the sampling distribution of means will be normal, and the normal distribution has been precisely defined such that it has a mean of 0 and a standard deviation of 1. We can use this information to compute the probability of a score occurring or the limits between which a certain percentage of scores fall

We discovered in Section 1.8.6 that 95% of z-scores fall between −1.96 and 1.96. This means that if our sample means were normally distributed with a mean of 0 and a standard error of 1, then the limits of our confidence interval would be −1.96 and +1.96.

the confidence interval can easily be calculated once the standard deviation (s in the equation) and mean ( in the equation) are known. However, we use the standard error and not the standard deviation because we’re interested in the variability of sample means, not the variability in observations within the sample.

As such, the mean is always in the center of the confidence interval. We know that 95% of confidence intervals contain the population mean, so we can assume this confidence interval contains the true mean; therefore, if the interval is small, the sample mean must be very close to the true mean. Conversely, if the confidence interval is very wide then the sample mean could be very different from the true mean, indicating that it is a bad representation of the population.

If we wanted to compute confidence intervals for a value other than 95% then we need to look up the value of z for the percentage that we want.

However, for small samples, the sampling distribution is not normal – it has a t-distribution. The t-distribution is a family of probability distributions that change shape as the sample size gets bigger (when the sample is very big, it has the shape of a normal distribution).

Confidence intervals provide us with information about a parameter, and, therefore, you often see them displayed on graphs.

the confidence interval tells us the limits within which the population mean is likely to fall. By comparing the confidence intervals of different means (or other parameters) we can get some idea about whether the means came from the same or different populations.

The fact that the confidence intervals overlap in this way tells us that these means could plausibly come from the same population: in both cases, if the intervals contain the true value of the mean (and they are constructed such that in 95% of studies they will), and both intervals overlap considerably, then they contain many similar values. It’s very plausible that the population values reflected by these intervals are similar or the same.

Analyze the data: fit a statistical model to the data – this model will test your original predictions. Assess this model to see whether it supports your initial predictions.

How do statistical models help us to test complex hypotheses

Null hypothesis significance testing (NHST) is a cumbersome name for an equally cumbersome process.

Fisher’s basic point was that you should calculate the probability of an event and evaluate this probability within the research context.

Neyman and Pearson believed that scientific statements should be split into testable hypotheses.