Multiple-choice example

Solution The mean and median would very RARELY have EXACTLY the same value. The median is indeed the middle score (the 50th percentile). The median is LESS vulnerable. We have our answer.

Example

Solution A grouping variable is a set of CODE NUMBERS, not real scores. Yes. It’s absolutely essential. We have our answer.

Last week How to enter data into SPSS. How to obtain various displays and graphs of the Caffeine data.

The SPSS Data Editor Works in two display modes: VARIABLE VIEW, in which you enter information about the variables in your data set; DATA VIEW, in which you enter your data.

Entering data into SPSS When entering data into SPSS, begin in Variable View. Good work in Variable View confers benefits both at the stage of data entry and when you are viewing the output.

Variable View completed Actually, value labels Variable View controls the appearance of Data View. Variable View also controls the appearance of the output. The graphs I produced with SPSS all had clear labels, for example. Variable View creates your WORKING ENVIRONMENT.

Between subjects experiments In the caffeine experiment, each of the participants in an experiment is tested under only ONE of the conditions making up the independent variable. In this experiment, the conditions making up the independent variable are said to vary BETWEEN SUBJECTS, and the experiment is said to be of BETWEEN SUBJECTS design.

Within subjects experiments In WITHIN SUBJECTS experiments, each participant is tested under ALL conditions. The Stroop experiment, which I described in the first lecture, was of within subjects design: each participant performed under all three conditions.

Grouping variables When entering data from a between subjects experiment, you need a GROUPING VARIABLE. A GROUPING VARIABLE is a column of code numbers or VALUES, each number representing the condition under which a score was achieved. We can let 1 = ‘Placebo’ and 2 = ‘Caffeine’, where 1 and 2 are VALUES and ‘Placebo’ and ‘Caffeine’ are VALUE LABELS.

Part of Data View In Data View, all the scores from the experiment appear as a SINGLE COLUMN of numbers. The Group variable appears as a single column of CODE NUMBERS.

Seeing the value labels To see the value labels in Data View (instead of the values),click Value Labels in the View menu. Seeing the value labels helps you avoid transcription errors when inputting data.

Graphs and displays We looked at two kinds of graphs: those that are pictures of DISTRIBUTIONS; those that SUMMARISE DISTRIBUTIONS by picturing the statistics.

Histograms and Stem-&-leaf displays Histograms and stem-and-leaf displays are pictures of DISTRIBUTIONS.

A histogram

Stem-and-leaf display

Bar charts and box plots BAR CHARTS and BOX PLOTS are pictures of the STATISTICS of distributions.

Bar chart (with error bars) standard deviations means

Types of bar chart SPSS offers three types of bar chart: Simple Clustered Stacked I do not like stacked bar charts, so I shall consider only the Simple and Clustered types.

Suppose there are two grouping variables in your data set We have run another caffeine experiment. But this time, we want to include the participant’s gender in the data set. We add a second grouping variable, Gender, where 1 = Male and 2 = Female. As well as comparing the performance of the Caffeine and Placebo groups, we also want to compare the mean scores of the female and male participants.

Two grouping variables

Clustered bar chart

Explanation Here the ‘category variable’ is Experimental Condition. The ‘Cluster variable’ is Gender. It looks as if the males are a little better under the Placebo condition. It looks as if the females are a little better under the Caffeine Condition.

Boxplots Extreme score Upper quartiles whiskers medians Lower quartiles Outlier

Lecture 6 Descriptive statistics with SPSS

The three most important properties of a distribution Its typical value, AVERAGE or CENTRAL TENDENCY, measured by the MEAN, the MEDIAN and the MODE. The SPREAD or DISPERSION of scores around the average value, measured by the STANDARD DEVIATION and RANGE STATISTICS such as the SIMPLE RANGE, the INTERQUARTILE and the SEMI-INTERQUARTILE RANGES. The SHAPE of the distribution.

Results of the caffeine experiment

Obtaining descriptive statistics Descriptive statistics such as the mean and standard deviation can be obtained through a variety of SPSS procedures. The MEANS procedure gives the means and standard deviations of data in different groups.

Finding the Means procedure

The Means procedure The Desciptive Statistics procedure does not allow comparison between groups and is inappropriate for grouped data. Our data set has a grouping variable. The Means option in Compare Means is the best one to start with. Note that the Descriptive Statistics procedure is only suitable for data sets that have no grouping variables. Descriptives would only calculate the statistics of ALL the scores.

The Means dialog box The variables in your data set are listed here initially. Select an item by clicking on it: it will appear in inverse video. The central arrows will come live and you can use one of them to transfer the item to the appropriate box. Click here to order additional statistics.

The Options dialog box To add extra items, click the first (to select it) then, holding the Control button down, click on the other items you want to include. Click the black arrow in the central pillar to transfer the statistics to the right hand panel. Click the Continue button to return to the Means dialog.

The SPSS Output Viewer

The output There are two tables. The Case Processing Summary warns you if SPSS has failed to include any data in the analysis. The Report is a table containing the statistics you asked for.

The Case Summary Clearly, SPSS found no problems with these data. All the data have been included in the analysis.

The Report The Placebo and Caffeine means are 9.25 and 11.90, respectively. The corresponding medians are 9.00 and 12.50. As we saw before, the simple range of values in either distribution is 15 hits.

Editing SPSS output If you don’t like a table in the Output, you can alter its appearance dramatically. You can remove information you do not need. You can change the shape of the table. The first move is to double-click on the image in the Viewer.

Changing the table Double-click on the item concerned, to produce a shaded border.

The Output Editor Click on Pivot to access the Pivot menu.

The Pivot menu

A more compact table

Leaving the Editor Click the File drop-down menu (in the Editor) and choose Close. Click anywhere in the Viewer outside the rectangle, which will then disappear.

Some revision

Percentiles A PERCENTILE is the VALUE or SCORE below which a specified percentage or proportion of the distribution lies. The 30th percentile is the value below which 30% of the distribution lies. The 70th percentile is the value below which 70% of scores lie.

The 30th and 70th percentiles (0.70) 0.30 30th percentile 0.70 (0.30) 70th percentile

The median is the 50th percentile 0.50 0.50 50th percentile (median)

The 2.5th and 97.5th percentiles The distribution of IQ has a mean of 100 and an SD of 15. 130 is 2 SD’s above the mean. Below that value lies 0.95 + 0.025 = 0.975 or 97.5% of the distribution. So 130 is the 97.5th percentile. Similarly the value 70 lies 2SD’s below the mean. Only .025 = 2.5% of values lie below that. An IQ of 70 is the 2.5th percentile. 0.95 (95%) 2 ½ % = .025 2 ½ % = .025 mean mean – 1.96×SD mean +1.96×SD

Populations and samples I measure the IQs of 20 people. The POPULATION is the reference set containing the reaction times of ALL people. The 20 IQs I have are merely a subset or SAMPLE from the population. Suppose I calculate their mean and SD. I might find that M = 108.0 and SD = 16.1.

Statistics versus parameters STATISTICS are characteristics of SAMPLES; PARAMETERS are characteristics of POPULATIONS. A normal population has two parameters: the mean; the standard deviation. The IQ population has a mean of 100 and an SD of 15.

The meaning of ‘error’ The population mean and standard devation are 100 and 15, respectively; whereas the sample mean and standard deviation are 108.0 and 16.1. We usually study samples in order to make inferences about the populations from which they have been drawn. But if the statistics of samples are taken as ESTIMATES of parameters, they are likely to be in error, to at least some extent.

Sampling variability Population characteristics are fixed. But the values of the corresponding statistics wobble around the values of the parameters. This ‘wobble’ is known as SAMPLING VARIABILITY or SAMPLING ERROR.

Sampling variability Suppose I take 25 people’s IQs and calculate the mean. It might be 95.1 . I do this 4000 times, calculating the value of the mean each time. I command SPSS to draw a histogram of the distribution of the MEANS of the 4000 samples.

Distribution of the sample mean

Distribution of the sample mean The values of the mean range from 89.9 to 110.0 . SAMPLING implies SAMPLING VARIABILITY or SAMPLING ERROR.

The laws of large numbers A set of theorems in probability theory. Their import is that if you make the sample size large enough, the mean of the sample will be as close as you wish to the value of the population mean. So if we take the mean and standard deviation of 4000 IQs, their values should be close to 100 and 15, respectively.

Distribution of 4000 IQs

The 2.5th and 97.5th percentiles Since this is a very large sample from a normal population, we should find that the 97.5th percentile is close to 130 (2 SDs above the mean) and the 2.5th percentile is close to 70 (2 SDs below the mean). To obtain these percentiles, we can use SPSS’s Freqencies procedure.

The Frequencies procedure It’s in the Descriptive Statistics menu. Click on Frequencies… to enter the Frequencies dialog box.

The Frequencies dialog Click on the Display frequency tables box to remove the arrow. This avoids a huge output of cumulative frequencies.

The output As you would expect, the mean is close to 100. The mean and median have similar values: 99.90 to 2 places of decimals. The 2.5th and 97.5th percentiles are close to 70 and 130, respectively.

Exercises I have made the SPSS data file for the 4000 IQs available on my Website. Command SPSS to draw a histogram of the distribution. Use the Descriptives procedure to find the mean and standard deviation of the distribution. I suggest you try the Frequencies procedure to obtain the 2.5th and 97.5th percentiles.

The caffeine data I have also brought the caffeine data on site. Try using the Means procedure to obtain descriptive statistics of the Caffeine and Placebo distributions. Try producing boxplots and a bar chart. Introduce a Gender grouping variable into the data and obtain a clustered bar chart.