# Study question: distribution of IQ

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Study question: distribution of IQ
The IQ has an approximately normal distribution, with a mean of 100 and a standard deviation of 15. If 1000 people are drawn at random from the population, how many of them can we expect to have IQs … greater than 130? between 100 and 130? less than 85?

Greater than 130? … If a variable is normally distributed, 95% of values lie within 1.96 standard deviations (2 approx.) on EITHER side of the mean. An IQ of 130 is TWO standard deviations above the mean of 100. 0.95 (95%) 2 ½ % = .025 2 ½ % = .025 mean mean – 1.96×SD mean +1.96×SD

Greater than 130? … Only 2 ½ per cent (0.025) of values lie more than 2 standard deviations above the mean. 2 ½ (2.5) per hundred is 25 in a thousand, which is our answer: about 25 people should have IQ’s greater than 130. 0.95 (95%) 2 ½ % = .025 2 ½ % = .025 mean mean – 1.96×SD mean +1.96×SD

Between 100 and 130? 95% of values lie within 2 standard deviations of the mean on either side. Half (47 ½ %) of these values lie above the mean. 47 ½ (47.5) in a hundred is 475 in a thousand, which is our answer. 0.95 (95%) 2 ½ % = .025 2 ½ % = .025 mean mean – 1.96×SD mean +1.96×SD

Less than 85? 68% of the distribution lies within 1 standard deviation on either side of the mean. 85 is ONE standard deviation below the mean. Half (50%) of the distribution lies below the mean. The area below 1SD below the mean (shaded) is (50 – 34) = 16%. That’s 160 in a thousand, which is our answer. 0.68 (68%) Mean – 1SD 34% 34% Mean + 1SD 16%

Study question At which percentile in the IQ distribution is
an IQ of 130? an IQ of 115? an IQ of 100? an IQ of 85?

130 130 is 2 SD’s above the mean. Below that value lies = or 97.5% of the distribution. So 130 is the 97.5th percentile. 0.95 (95%) 2 ½ % = .025 2 ½ % = .025 mean mean – 1.96×SD mean +1.96×SD

115 115 is ONE SD above the mean. From the diagram, ( ) = 84% of the distribution lies below 115, which is, therefore the 84th percentile. 0.68 (68%) Mean – 1SD 34% 34% Mean + 1SD 16%

100 A normal distribution is centred on the mean.
So 50% of observations lie below the mean. 100 is the 50th percentile of the IQ distribution. 50% 100 (mean)

85 An IQ of 85 is one SD below the mean.
The area below that is 16%, so 85 is the 16th percentile of the distribution. 0.68 (68%) Mean – 1SD 34% 34% Mean + 1SD 16%

Lecture 5 Graphs 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.

Get to know your data Statistics such as the mean can be misleading.
The dispersion and shape of the distributions can tell a more accurate story of what happened during the experiment. Ceiling and floor effects can mask the action of the independent variable. Outliers can exert undue leverage upon the values of some statistics.

Results of the caffeine experiment

Getting to know your data with SPSS
We shall now use SPSS to obtain pictures of this data set as a whole. We shall also obtain descriptive statistics of the Caffeine and Placebo distributions.

The opening SPSS dialog

The opening dialog … Click the ‘Type in data’ radio button at one down from top left. Click the OK button at the bottom. This will get you into the Data Editor, which you can see in the background. Click this button

The data editor (Data View)

Data View You have been looking at Data View, one of the Data Editor’s two displays. The other display is Variable View, which we shall look at in a moment. You could begin to enter data into the grid immediately. DON’T DO THAT: the format will be horrible. Always begin in Variable View, which is accessed by clicking on a tab at the bottom of Data View.

The data editor (Data View)

Variable View

Variable View Variable View controls the appearance of everything in Data View and much else besides. It NAMES the variables. It controls the FORMAT of the numbers. It controls aspects of the appearance of the OUTPUT. It gives SPSS essential information about the nature of your data.

Variable View … Variable View sets up the working environment you will experience in Data View.

Variable View The name that will appear in Data View
The name that will appear in the output. This contains the key to any code numbers. Level of measurement Click tab to enter Data View Each row in Variable View contains information about ONE of the variables in your data set.

Levels of measurement SPSS classifies data according to the LEVEL OF MEASUREMENT. There are 3 levels: SCALE data, which are measures on an independent scale with units. Heights, weights, performance scores, counts and IQs are scale data. Each score has ‘stand-alone’ meaning. ORDINAL data, which are ranks. A rank has meaning only in relation to the other individuals in the sample. It does not express the degree to which a property is possessed. NOMINAL data, which are assignments to categories. (So-many males, so-many females.)

Graphics The latest SPSS graphics require you to specify the level of measurement of the data on each variable. The group code numbers are at the NOMINAL level of measurement, because they are merely CATEGORY LABELS. Make the appropriate entry in the Measure column.

Places of decimals Note the Decimals column.
You don’t want whole numbers such as 1 or 2 appearing in Data View as 1.00 and 2.00 – that’s too cluttered. You can fix that while you are still in Variable View by making an entry of zero in Decimals. When you move to Data view, the numbers will appear as 1 and 2.

SPSS data sets First, we have to rearrange the results of our caffeine experiment into a form that SPSS will accept. Our table of data is NOT acceptable to SPSS. EACH ROW must contain data on just ONE participant. Each COLUMN must represent a VARIABLE. This row contains all the data on Participant 6. This column contains all the data on ONE variable

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.

A within subjects experiment
Here are the results of an experiment in which each participant tries to recognise words presented in the right and left visual fields. Here the conditions making up the independent variable are said to vary WITHIN SUBJECTS, and the experiment is said to be of WITHIN SUBJECTS design.

A grouping variable In the caffeine experiment, there were two groups of participants. We need to inform SPSS of each participant’s group membership by including a GROUPING VARIABLE in the dataset. A GROUPING VARIABLE is a set of code numbers or VALUES, each number representing the condition under which a score in the same row was achieved. We can let 1 = ‘Placebo’ and 2 = ‘Caffeine’, where 1 and 2 are VALUES and ‘Placebo’ and ‘Caffeine’ are VALUE LABELS.

Specifying the level of measurement
The code numbers of the grouping variable are merely LABELS. A grouping level is at the NOMINAL level of measurement.

Variable View completed
Actually, value labels The (variable) NAME is what will appear in Data View. The (variable) LABEL will appear in the output. Adjust the Decimals to zero – avoid clutter. You only need Value (labels) for the grouping variable, not for the scores themselves.

Assigning value labels

In Variable View … The name must be a continuous string of letters (or letters and numbers): no spaces are allowed. Preserve phrasing by using upper and lower case: ‘TimeOfDay’.

Variable labels The ‘Label’ is a proper caption, complete with spacing: Time of Day. The label appears in the output. The label will not appear in Data View. Careful choice of labels greatly improves the intelligibility of SPSS output.

Grouping variables are only needed when you are analysing data from between subjects experiments

Decimals The Decimals column controls the number of places of decimals of values displayed in Data View. By default, numbers will be displayed to two decimal places. So 2 will appear as 2.00. Click on Decimals and reset the value to zero to display whole numbers, with no decimal point.

Part of Data View Note that variable ‘Group’ consists of code numbers identifying the conditions under which the score in the same row was achieved. There are no decimals.

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.

We need better graphs This sort of diagram only works when you have a variable with a few different values and a small data set. For larger data sets, we need different graphs and displays.

A histogram

Finding the histogram command
The command for a histogram is found in the Graphs menu. Histograms are also obtainable on other SPSS menus.

The histogram dialog Just click to select (highlight) the variable in the left panel whose distribution you want to graph. Click the top arrow to transfer the variable to the ‘Variable’ slot at top middle. Click the OK button to obtain your histogram.

Stem-and-leaf display

Stem-and-leaf display
The range is stepped out on a vertical ‘stem’. The individual observations are the ‘leaves’. As with the histogram, you see the shape of the distribution. And you can at (least partially) recover the original data.

Finding the stem-and-leaf display
The stem-and-leaf display is an option in Explore, which can be found in Descriptive Statistics in the Analyze menu. Just click on Explore to obtain the stem-and-leaf dialog.

Finding the stem-and-leaf display
The name of the variable whose distribution you want to display goes in here. Click the Stem-and-leaf check-box. Click on Plots… to enter the Explore:Plots dialog.

Graphs that summarise distributions
The histogram and stem-and-leaf display are pictures of a distribution. Sometimes we shall want to have a picture that allows us to compare SUMMARIES of distributions. The BAR CHART is such a graph.

Bar chart (with error bars)

Bar chart … The heights of the bars represent the group means.
The ERROR BARS represent standard deviations The heights of the bars represent the group means. The thin ERROR BARS represent the standard deviations of the scores (or related statistics). A bar chart does not show the SHAPE of the original distribution. Bar charts are found in the Graphs menu. This is what is known as a SIMPLE BAR CHART.

Finding bar charts In the Graphs menu, click on Bar … to enter the Bar Charts dialog box.

The Bar Charts dialog We want the Simple Bar Chart
We want to summarise the scores of the Caffeine and Placebo groups.

Completing the dialogs
The heights of the bars will be proportional to the group means Click Display error bars box. Click Options to include error bars The error bars will represent the standard deviations

The simple bar chart

Boxplots… In the Explore:Plots dialog, we can also order Boxplots.
Unlike bar charts, boxplots can tell us something about the SHAPE of the distribution.

Boxplots of the Placebo and Caffeine distributions
Extreme score Upper quartiles whiskers medians Lower quartiles Outlier

Boxplots… A skewed distribution would be indicated by a median line that was closer to one end of the box than the other. As we know, that is not the case with the data from the Caffeine experiment.

Summary When entering data into SPSS, start in Variable View first.
Good work in Variable View confers benefits both at the stage of data entry and when you are viewing the output. We looked at two kinds of graphs: those that depict DISTRIBUTIONS those that SUMMARISE DISTRIBUTIONS by picturing the statistics

Summary… Histograms and stem-and-leaf displays are pictures of distributions. Bar graphs and boxplots summarise distributions by picturing their statistics.

‘Getting started with SPSS 14’
Kinnear & Gray, Chapter 2 Today’s topic is covered in more detail in Chapter 2 of the recommended textbook. The title of the chapter is ‘Getting started with SPSS 14’ Chapter 5 has more on graphics.

Multiple-choice example

Another example