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QPSMR formulae

This document describes the calculations used by QPSMR when calculating table based statistics and significance levels (probabilities).

Figures included in the calculations

Most of the statistics are based on a single column or compare columns of a table using the rows as follows:

Statistics only. The table accumulates the necessary figures from an integer or weight (floating point) value. The accumulated values are used for the calculations.  Any respondents with "Undefined" values are excluded altogether.

List all rows. The rows are a list of all the values found from an integer or weight (floating point) value.&nbspThe values are used for the calculations. Any respondents with "Undefined" values are excluded altogether.

Score values. The rows are single-coded or multi-coded responses with score values assigned to them. The score values are used for the calculations. Any respondents with no score value attached (for example DK/NA rows) are excluded altogether.

The rows used for the calculations can be suppressed (format NDIS).

Some statistics (formats MED, ILE, ILL, ILH) cannot be produced from the accumulated figures in a “Statistics only” table: a “List all rows” table is needed.  Format RNA should also be used.

All the formulae use the real figures calculated for the table, not the printed figures that may have been rounded.

Weighted data

Weighting the data affects the calculations.  There are two sorts of weighting in QPSMR:

Respondent weighting (for example target or rim weighting). Each respondent is given a “weight” to adjust the figures to some known population or to compensate for sample imbalances.

Quantity weighting.  A value is used to weight the table so that the figures in the table are not respondents, but represent the total of some value from the questionnaire, for example total area or volume.

Both types of weighting can be applied on the same table.

The calculations generally use the weighted figures.

Where appropriate the un-weighted or effective number of respondents is used.  This is shown as Ne in the formulae.

If the recommended format ESS is used then the effective sample size used in calculations is:

Where w is the weight applied to each respondent.

If ESS has not been set, then Ne is the un-weighted number of respondents.

Significance markers

When marking significance levels against or under the figures the markers are:

Asterisks (stars). These show a comparison with the total column.  The two columns that are compared are the test column and the total column minus the test column.  The test for a column is therefore a test against the rest of data. The markers are used when the column is higher or lower than the rest of the data. One star shows the lower significance level and two stars show the higher level.

Letters. The column letters are shown under the column labels.   The columns are normally compared one at a time with all the other columns under the same heading (format SHG1).  The letter markers are only placed on the column with the higher value. If the column heading letters are lower case then a lower case letter shows the lower significance level and an upper case letter shows the higher level.

Additional higher levels of significance can be shown with a plus sign (+), or even higher with two plus signs (++).

Mean score or average


x is each value or score value.

n is the (weighted) number of respondents with that value.

N is the (weighted) base (sum of n).

Standard deviation

This is calculated as:

This is normally written as:


This is calculated as:

This is normally written as:

Standard error

Error variance (sample variance)

Mean over standard error

Used when the expected mean is zero:

Means comparison t-test

When comparing two column means with TTV1:

With TTV2:

Proportions comparison Z test

When comparing two column percentages in the same row.  Where:

p is each proportion

pt is the combined proportion from both columns added together

N is the base

When comparing proportions (percentages) with SIG1:


With SIG2:

F-test formula


F = between MSS / within MSS


between MSS =  adjusted between SS  / ( ncols - 1 )

within MSS =  adjusted within SS  / ( un-weighted total - ncols )


adjusted between SS = between SS  ( un-weighted total  /  weighted total )

adjusted within SS = within SS  ( un-weighted total  /  weighted total )


ncols = number of table columns used for F test

un-weighted total = un-weighted total of respondents used in calculation

weighted total = weighted total of respondents used in calculation