Summarising data / Levels of measurement / Introduction to SPSS Topic 2.

Summarising data /

Levels of measurement /

Introduction to SPSS

Topic 2

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Main Issues for this session

Levels of measurementData types: nominal, ordinal, interval, ratio

Linking data types to statistical analyses Introduction to SPSS

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Reading

Chapter 2 and Chapter 3Frequency Distributions and Graphic

Representation

Fundamentals of Statistical Reasoning in Education,

Colardarci et al.

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Levels of measurement

Sequential Magnitude Zero point ExampleDescriptive Statistics

NominalPercent, ratio,

frequencyOrdinal x

Interval x x ArbitraryMean, SD, Min,

MaxRatio x x Absolute

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Preparing a questionnaire and codebook Example questionnaire: Example codebook: Example codebooks:

http://pisa2006.acer.edu.au/downloads.php

WB_Pupil_MP.doc

Pupil_codebooks.xls

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Codebook - 1

A codebook should be prepared as a questionnaire is developed

The purposes of a codebook areTo facilitate data entry, with codes shown on

the questionnaire if possibleTo plan for analysis; to help with determining

the types of analyses that are appropriate.

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Codebook - 2

Numeric codes are easier to enter than alphabetic codes

Consider the appropriate field width and range of answers. These can be useful feedback to questionnaire design as well.

Decide how to handle missing responses

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Getting data into SPSS - 1

The EXCEL file contains the pupil questionnaire data

Import this data set into SPSS: Start SPSS

Puipl_data.xls

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Select from MenuFile -> Open -> Data

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Getting data into SPSS - 3 Find the folder where the EXCEL file is stored. In the file open dialog box, make sure the file type

is set to xls.

File type set to “xls”

Select file Pupil_data.xls

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Make sure the check box for “Read variable names from the first row of data” is checked. (The EXCEL file has variable names in the first row, and these will be read in as SPSS variable names as well.

Check this box

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Toggle between data view and variable view The tab at the bottom left corner shows

the data view or variable view.

Data view or Variable view

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Add Variable labels for variables 4 to 9 (PDOBDD to PHOMLANG)

Variable label

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Add Value labels for variable PSEX

(The column after Variable Labels). Click in the value labels cell and the

following dialog box appears

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Add missing values for variable PSEX (The column after Value Labels) Click in the Missing values cell and a dialog box

appears. Enter values representing missing values

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Practice for other variables

Set variable labels, value labels and missing values for some other variables

Copy and pasting value labels and missing values from a set of cells to other cells can be done.

Make sure you save the file often!!

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Frequencies

For which types of variables, will it be appropriate to compute frequencies? Nominal, ordinal, interval and ratio?

For which types of variables, will it be appropriate to compute averages? Nominal, ordinal, interval and ratio?

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Compute frequencies in SPSS -1

Select from menu Analyze -> Descriptive Statistics ->

Frequencies

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Select the variables in the left-hand box and move them to the right-hand box.

Press OK.

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Explore the options under the Statistics and Charts buttons, and see what kinds of output you can produce.

Compute frequencies for other variables as a practice.

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Constructs in a questionnaire - 1

Sometimes we are interested in a measure that is not directly obtainable/observable as questions like “are you a boy or a girl”.

For example, socio-economic status is something that we have an interest in, but it is a concept (like well-being) rather than something that we can see and directly measure.

Such concepts are often called constructs, or latent variables.

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Sociologists and statisticians have developed methodologies to measure constructs (or latent variables).

Psychometrics is the science of the measurement of latent variables.

The field of psychometrics include classical test theory (CTT) and item response theory (IRT)

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To measure a construct, typically a number of observable indicators are collected (e.g., through a questionnaire).

The data from these indicators are aggregated in some way (e.g., to form a total score) to be used as a measure of the construct for each individual.

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A simple way to aggregate the indicators into a measure for a construct is just to sum the scores for the set of questions for each student.

These sums (or measures of the constructs) can then be used as new variables as the basis of further statistical analysis.

There are more sophisticated ways to aggregate the indicator scores into a construct score (e.g, using item response theory models).

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In SPSS, calculate sum scores for each construct you identified, for each student.

You can then use these new variables for further analyses.

Watch animated demo on how to compute sum scores.

HowToComputeSumScores_demo.swf

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Outline

Categorical variables (ordinal and nominal)

Continuous variables (interval and ratio)

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Download from subject website Data file from TIMSS 2003 study for

Australia

TIMSS2003AUS.sav Student Questionnaire from TIMSS 2003

study for Australia

T03_Student_8.pdf

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Categorical data

Nominal - numbers are used only as labels for different objects within a set. For example, gender idbook (there are 12 different test booklets)

Ordinal - numbers are used to reflect the rank order of objects within a set according to a specific criterion bsbgbook (number of books in the home) bsbgmfed (mother’s education level)

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Summary of categorical variables In general, summary of categorical variables addresses

the questions: How many categories? How many cases in each category or What are the proportions of

cases in each of the categories? If a variable is ordinal, questions regarding trends and

association can be considered.

Examples: For data file TIMSS2003AUS.sav, the possible questions

could be: What are the proportions of female and male students in the

study? What are the levels of education of parents for the students

surveyed? Is there an association between levels of education of parents

and number of books in the home?

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Hands-on (1)

Are there more girls than boys? Is there an association between Father’s

education level and the number of books at home?

Follow animated demo frequency_1_demo frequency_2_demoExplore_1_demoExplore_1_output_demo

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Hands-on (2)

Is there a difference between girls and boys in terms of whether they enjoy mathematics (variable bsbmtenj)?

Follow animated demo Crosstab_1_demo Crosstab_1_output_demo

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Hands-on (3)

Is there a difference between girls and boys in terms of whether they enjoy SCIENCE (variable bsbstenj, (var 67))?

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Things to watch out for in comparing frequencies - 1 Consider if you should compare raw

frequencies or percentages. For percentages, make sure the

denominator (total) is the appropriate one to use. For example, check row total, column total, overall total.

Check the scale to make sure there is no exaggeration of differences

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Things to watch out for in comparing frequencies – Raw score or percentage?

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Things to watch out for in comparing frequencies – Raw score or percentage?

Percentages are better because there are many more students speaking the test language at home than those who do not.

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Things to watch out for in comparing frequencies – Check magnitude of scale

The graph on the right shows large differences. But check the scale on the vertical axis. There are only a few students. We can’t say there is a great difference.

Beware of visual deception.

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Continuous data

Interval - numbers reflect both the rank order of objects and the extent of the differences between them (e.g. temperature)

Ratio - scale has an absolute zero and hence a ratio of scores is independent of the units of the scale (e.g. height, weight, age. )

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Summary of continuous variables

Example of Questions

1. What is the average score that the students surveyed get?

2. What is the middle score? (median)3. Which is the most frequent score? (mode)4. What is the highest score ? (max)5. What is the lowest score? (min)6. What is the range of students’ scores?

(range)7. To what extent are the scores close to the

mean? (variance and standard deviation)

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Mean and Median

Mean (average, expected value)Sum observations / number of observations

Median50% subjects below and 50% subjects above

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Variance and Standard deviation

2

variance1

i

i

x

n

standard deviation= variance

Where µ is the mean, and n is the number of observations.

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Normal Distribution

Many variables have a distribution shaped like a bell curve.

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Example descriptive statistics

Variable 154 (bsmmat01) is an estimate of a student’s mathematics achievement.

Follow animated demo:descriptive_1_demo

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Histogram of continuous variable

Frequency analysis and bar charts may fail because there are too many categories.

Use histogram. Variable 154 (bsmmat01) is an estimate of

a student’s mathematics achievement. Follow animated demo:

histogram_1_demo

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Compare histograms for groups

Compare mathematics achievement distributions between groups based on father’s education level.

Follow animated demo:histogram_2_demo

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Box-Plots Box-plots are graphical representations of the

data in a five-number summary with the addition of ‘cutoffs’ or ‘fences’ for the identification of possible outliers (individual data points are plotted beyond the fences if they occur)

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Box plot for mathematics achievement Follow animated demo:

boxplot_1_demoboxplot_2_demo

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Output of Box-plot of mathematics scores

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Output of Box-plot of mathematics scores by father’s education level

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Parametric and Non-parametric

Mean and Median

Mean: average Median:

score at the 50th percentile.The middle value

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Mean and Median If the distribution of scores is symmetrical,

the mean and median will be close. If the distribution is skewed, then the mean

and median will be quite different. Mean is sensitive to outliers Median is not sensitive to outliers Example: income distribution

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Examples of income distribution

What will be the mean? What will be the median?

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Robust statistics

The mean will be much higher than the median, because there are four people with very high salaries.

The median will not shift if the four highest salaries are in the 150K range instead of the 280 range, but the mean will change by a great deal.

The median is said to be “robust”.

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Percentile Rank The percentile rank of a raw score s, is

the percentage of people whose scores are less than or equal to s. Example:

Raw 12 14 28 34 47 50

Rank 1 2 3 4 5 6

%Rank

1/6 2/6 3/6 4/6 5/6 6/6

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Advantages and disadvantage of percentile ranks Simple to communicate. More “robust” (not affected by extreme

scores in the distribution) Raw scores turned into Ranks:

reduce raw scores to ordinal measurement. Percentile ranks have uniform distribution,

not normal. Percentile differences in the middle of the

score range can exaggerate small differences.

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Compute percentile ranks in SPSS

Compute percentile ranks using mathematics achievement score

Follow animated demo: percentile_1_demo

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Histogram of percentile ranks

Do a histogram of percentile ranks, what do you see?

Plot a scatter graph of mathematics achievement (variable 154) with the newly created variable of percentile ranks.

How do you interpret the graph?

Summarising data / Levels of measurement / Introduction to SPSS Topic 2.

Documents

data slide

spss slide

box slide

variable view slide

data view

data set

variable labels

phomlang variable label