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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 …

1. greater than 130?2. between 100 and 130?3. less than 85?

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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%)

mean

mean – 1.96×SD mean +1.96×SD

2 ½ % = .025 2 ½ % = .025

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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%)

mean

mean – 1.96×SD mean +1.96×SD

2 ½ % = .025 2 ½ % = .025

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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%)

mean

mean – 1.96×SD mean +1.96×SD

2 ½ % = .025 2 ½ % = .025

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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%)

34% 34%Mean – 1SD Mean + 1SD

16%

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Study question

At which percentile in the IQ distribution is

1. an IQ of 130?

2. an IQ of 115?

3. an IQ of 100?

4. an IQ of 85?

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130

• 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.

0.95 (95%)

mean

mean – 1.96×SD mean +1.96×SD

2 ½ % = .025 2 ½ % = .025

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115

• 115 is ONE SD above the mean.

• From the diagram, (68 + 16) = 84% of the distribution lies below 115, which is, therefore the 84th percentile.

0.68

(68%)

34% 34%Mean – 1SD Mean + 1SD

16%

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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. 100

(mean)

50%

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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%)

34% 34%Mean – 1SD Mean + 1SD

16%

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Lecture 5

Graphs with SPSS

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The three most important properties of a distribution

1. Its typical value, AVERAGE or CENTRAL TENDENCY, measured by the MEAN, the MEDIAN and the MODE.

2. 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.

3. The SHAPE of the distribution.

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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.

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Results of the caffeine experiment

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Getting to know your datawith 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.

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The opening SPSS dialog

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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

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The data editor (Data View)

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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.

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The data editor (Data View)

Click here to enter Variable

View.

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Variable View

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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.

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Variable View …

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

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Variable View

Each row in Variable View contains information about ONE of the variables in your data set.

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

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Levels of measurement

SPSS classifies data according to the LEVEL OF MEASUREMENT. There are 3 levels:

1. 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.

2. 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.

3. NOMINAL data, which are assignments to categories. (So-many males, so-many females.)

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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.

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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.

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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

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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.

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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.

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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.

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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.

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Variable View completed

• 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.

Actually, value labels

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Assigning value labels

Click here to enter the Value Labels dialog.

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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’.

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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.

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Grouping variables are only needed when you are analysing data from

between subjects experiments

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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.

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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.

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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.

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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.

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A histogram

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Finding the histogram command

• The command for a histogram is found in the Graphs menu.

• Histograms are also obtainable on other SPSS menus.

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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.

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Stem-and-leaf display

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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.

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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.

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Finding the stem-and-leaf display

Click on Plots… to enter the Explore:Plots dialog.

Click the Stem-and-leaf check-box.

The name of the variable whose distribution you want to display goes in here.

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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.

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Bar chart (with error bars)

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Bar chart …

• 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.

The ERROR BARS represent standard deviations

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Finding bar charts

• In the Graphs menu, click on Bar … to enter the Bar Charts dialog box.

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The Bar Charts dialog

We want the Simple Bar Chart

We want to summarise the scores of the Caffeine and Placebo groups.

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Completing the dialogs

The error bars will represent the standard deviations

The heights of the bars will be proportional to the group means

Click Options to include error bars

Click Display error bars box.

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The simple bar chart

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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.

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Boxplots of the Placebo and Caffeine distributions

medians

Upper quartiles

Lower quartiles

Extreme score

Outlier

whiskers

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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.

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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:

1. those that depict DISTRIBUTIONS

2. those that SUMMARISE DISTRIBUTIONS by picturing the statistics

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Summary…

• Histograms and stem-and-leaf displays are pictures of distributions.

• Bar graphs and boxplots summarise distributions by picturing their statistics.

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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.

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Multiple-choice example

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Another example