The data collected by a researcher should first of all be described. Which measures are used for this description depends on how the measurements were made, i.e. on the so-called measurement scale of the variable.

Measurement scales

Correctly determining the type of analysis to be performed depends on the scale on which the collected data are expressed. There are 3 main measurement scales:

1. Interval scale (ang. interval scale)
   A variable is expressed on an interval scale when:

   • it can be ordered,
   • it is possible to calculate by how much one element is greater than another, and this difference has a real-world interpretation. A unit of measurement is usually defined.

   Example: the mass of an object [kg], the area of an object [m], time [years], speed [km/h], etc.

2. Ordinal scale (ang. ordinal scale)
   A variable is expressed on an ordinal scale when:

   • it can be ordered, i.e. the order in which the elements occur matters,
   • it is not meaningful to determine the difference or the ratio between two values.

   Example: education level, the order of competitors on a podium, etc.
   

   Note! If a variable is expressed on an ordinal scale, then, for calculations to be performed on it correctly, it should be recorded using numbers. These numbers are conventional identifiers denoting the order of the elements.

3. Nominal scale (ang. nominal scale)
   A variable is expressed on a nominal scale when:

   • it cannot be ordered, i.e. there is no ordering that follows from the nature of the phenomenon,
   • it is not meaningful to determine the difference or the ratio between two values.

   Example: sex, country of residence, etc.
   

   Note! If a variable is expressed on a nominal scale, it may be recorded using text labels. Even if the values of a nominal variable are expressed numerically, these numbers are only conventional identifiers – so no arithmetic operations may be performed on them, nor may they be compared.

Before starting an analysis, it is recommended to assign a measurement scale to each variable. Doing so will give the variable headers a colour corresponding to that scale, i.e. green = interval scale, yellow = ordinal scale, red = nominal scale. The colour of the variables (and thus their scale) will be visible in the data sheet and in the list of variables in the analysis windows.
A scale can be assigned to a selected variable in the variable options window Codes/Labels/Format or from the context menu on the header of the selected variable.

Quantitative data comprise any information that can be quantified, counted or measured and given a numerical value. Quantitative data include the interval scale and, sometimes, also the ordinal scale.
Qualitative data are descriptive in nature, expressed in linguistic categories rather than numerical values. Qualitative data include the nominal scale and, sometimes, also the ordinal scale.
An ordinal scale that has many objectified categories is quantitative data, e.g. the SF-36 quality-of-life scale (0 to 100 points). If, however, there are so few categories that they can be described in words, e.g. education (primary, secondary, higher), it is qualitative data.

Tables

Frequency tables and empirical distributions of data

When to use:

The foundation of statistical research is determining the empirical distribution, i.e. the distribution of a feature observed in the sample – in other words, checking how often the individual values of the feature recur. Such a distribution is presented in the form of a frequency table or a chart (histogram). For small data sets, frequency tables can present all the data – the so-called point frequency distribution; for larger data sets, so-called interval (class) frequency distributions are created instead.

How to set up the analysis

Statistics \(\to\) Descriptive analyses \(\to\) Frequency tables.

In this window we select the variable to analyse and the analysis options. Depending on the options chosen, the returned result can be sorted treating the variable's values as text or as numbers. If there are empty cells in the analysed column, they may be included in the analysis or omitted. The analysis result appears in the report attached to the data sheet for which it was performed.
If, in addition, we want the data to be illustrated with a column chart or a histogram, we select the Add graph option in the Frequency tables window.

Performing and interpreting the analysis

Table report

When to use:

The table report makes it possible to summarise a large amount of data simultaneously in the form of two-way tables (tables of two features). For example, in this way we can present the distribution of age groups by place of residence, education, and so on. Each table is presented as counts within particular categories and can additionally be summarised by calculating row percentages, column percentages, or percentages of the grand total, and it is also possible to determine the table of expected frequencies. For such tables an automatic summary in the form of a column chart is also available.

How to set up the analysis

Statistics \(\to\) Descriptive analyses \(\to\) Table report.

Performing and interpreting the analysis

Analyses for contingency tables

When to use:

Analyses for contingency tables are used when we want to examine the relationship between two (or more) qualitative features – e.g. whether the type of medication taken is associated with the occurrence of complications. They are calculated on the basis of data collected in a contingency table or directly from data in raw form. It is also possible to transform data from a contingency table into raw form, or vice versa.

Basic tests for contingency tables:

• Chi-square goodness-of-fit test
• Chi-square test for RxC (2x2) tables and its corrections
• Chi-square trend test for Rx2 tables
• McNemar's test, Bowker's test of internal symmetry
• Chi-square test for multidimensional contingency tables
• Cochran's Q ANOVA
• The Mantel-Haenszel method for multiple 2x2 tables

Coefficients for contingency tables:

• Relative Risk and Odds Ratio
• Contingency coefficients: Yule's Q, Phi, Cramer's V, Pearson's C
• Cohen's Kappa coefficient of agreement
• Fleiss' Kappa coefficient
• Sensitivity and specificity, PPV, NPV, LR(+), LR(-), prevalence, accuracy

Definitions of measures

The results report can also include a basic summary of the tables:

• Contingency table of observed frequencies \(-\) i.e. data in the form of a contingency table. Such a table shows the distribution of observations for several features (several variables). A table for 2 features (\(X\), \(Y\)), of which the first has \(r\) possible categories and the second \(c\), is shown below (table 1).

  Table 1. The \(r\times c\) contingency table of observed frequencies

  Frequencies		Feature Y
  observed \(O_{ij}\)		\(Y_1\)	\(Y_2\)	...	\(Y_c\)	Total
  Feature \(X\)	\(X_1\)	\(O_{11}\)	\(O_{12}\)	...	\(O_{1c}\)	\(\sum_{j=1}^cO_{1j}\)
  	\(X_2\)	\(O_{21}\)	\(O_{22}\)	...	\(O_{2c}\)	\(\sum_{j=1}^cO_{2j}\)
  	...	...	...	...	...	...
  	\(X_r\)	\(O_{r1}\)	\(O_{r2}\)	...	\(O_{rc}\)	\(\sum_{j=1}^cO_{rj}\)
  	Total	\(\sum_{i=1}^rO_{i1}\)	\(\sum_{i=1}^rO_{i2}\)	...	\(\sum_{i=1}^rO_{ic}\)	\(n=\sum_{i=1}^r\sum_{j=1}^cO_{ij}\)

  Observed frequencies \(O_{ij}\) (\(i=1,2,\dots,r;\quad j=1,2,\dots,c\)) show the frequency of occurrence of the individual categories for both features.
  For such a table to be returned by the program, the option Include analysed data must be selected in the test window. For the data from example (4), the contingency table of observed frequencies is as follows:

  

• Contingency table of expected frequencies \(-\) for each contingency table of observed frequencies, a corresponding table of expected frequencies: \(E_{ij}\) can be created (table 2) – i.e. the frequencies we would expect in the individual cells of the table if both features were independent of each other (formula in For those interested).

  Table 2. The \(r\times c\) contingency table of expected frequencies

  Frequencies		Feature Y
  expected \(E_{ij}\)		\(Y_1\)	\(Y_2\)	...	\(Y_c\)
  Feature \(X\)	\(X_1\)	\(E_{11}\)	\(E_{12}\)	...	\(E_{1c}\)
  	\(X_2\)	\(E_{21}\)	\(E_{22}\)	...	\(E_{2c}\)
  	...	...	...	...	...
  	\(X_r\)	\(E_{r1}\)	\(E_{r2}\)	...	\(E_{rc}\)

  For the data from example (4), the contingency table of expected frequencies is as follows:

  

• Contingency table of percentages calculated from the column totals. For the data from example (4), this table is as follows:

  

• Contingency table of percentages calculated from the row totals. For the data from example (4), this table is as follows:

  

• Contingency table of percentages calculated from the grand total of rows and columns. For the data from example (4), this table is as follows:

  

Descriptive statistics

When to use:

The purpose of descriptive statistics methods is to summarise a data set using a few numerical characteristics, e.g. the mean value, the median, or the standard deviation, and to draw basic conclusions and generalisations about the data set on that basis – without this step it is difficult even to begin to understand the data we are dealing with.

How to set up the analysis

Statistics \(\to\) Descriptive analyses \(\to\) Descriptive statistics.

In this window we select the variable to analyse and the analysis options, and check the descriptive statistics measures we are interested in. Individual statistics or whole groups of them can be selected using the button. We confirm the selection with the OK button. The analysis result appears in the report attached to the data sheet for which it was performed.
If, in addition, we want the data to be illustrated with a box-and-whisker plot, we check the Add graph option in the Descriptive statistics window.

Measures of central tendency

Measures of central tendency are so-called average measures, characterising the mean or typical level of the values of a feature.

Definitions of measures

The arithmetic mean (ang. arithmetic mean) is the best known of the average measures – the sum of all the values divided by their number (formula in For those interested). It is used for the interval scale. For a sample it is denoted by \(\overline{x}\), and for a population by \(\mu\).
The trimmed mean – computed as the arithmetic mean after removing a given percentage of the smallest and largest measurements from the sample, e.g. trimming 5% of the measurements means removing 2.5% of the largest and 2.5% of the smallest values. If the number of measurements to be removed, obtained from this calculation, is not a whole number, it is rounded down to the nearest integer.
The Winsorised mean – computed as the arithmetic mean after replacing the corresponding percentage of extreme measurements with the smallest and largest value remaining in the reduced set of values. If we decide to compute the Winsorised mean by trimming, say, 5% of the measurements, the discarded 5% is replaced by the smallest and largest value determined from the remaining 95% of the measurements. As with the trimmed mean, if converting the percentage of values to be replaced does not give a whole number, it is rounded down to the nearest integer.
The geometric mean (ang. geometric mean) is the \(n\)th root of the product of all the values (formula in For those interested). It is used for the interval scale when the variable has a log-normal distribution (the logarithm of the variable has a normal distribution).
The harmonic mean (ang. harmonic mean) is the number of observations divided by the sum of the reciprocals of the individual values (formula in For those interested). It is used for the interval scale.
The median (ang. median)
In an ordered data set, the median is the value that divides the set into two equal parts. Half of all the observations lie below, and half above, the median.

The median can be used for the interval scale and the ordinal scale.
The mode (ang. mode)
The mode \(-\) is the value that occurs most frequently among the measurements obtained. The mode can be used for any measurement scale.

Other measures of location

Quartiles (ang. quartiles), deciles (ang. deciles), centiles (ang. centiles) are values that divide an ordered data set into equal parts – similarly to the median, but into more than two parts.

Quartiles (\(Q_1\), \(Q_2\), \(Q_3\)) divide an ordered series into 4 equal parts, deciles (\(D_i\), \(i=1,2,...,9\)) into 10 equal parts, and centiles (percentiles: \(C_i\), \(i=1,2,...,99\)) into 100 equal parts. The second quartile, the fifth decile and the fiftieth centile are all equal to the median. These measures can be used for the interval scale and the ordinal scale.

Measures of dispersion

Knowledge of measures of central tendency is not enough to characterise the structure of a statistical collection – the group under study may show varying degrees of variability in the feature examined. Measures are therefore needed that allow us to determine the spread of the data.

Assumptions

Definitions of measures

The range (ang. range) – the difference between the largest and smallest value in the sample (formula in For those interested). The interquartile range (\(IQR\)) is the analogous difference between the upper and lower quartile.
Ranges for the percentile (decile, centile) scale. Ranges between percentiles are one of the measures of dispersion – they indicate the percentage of all observations whose value lies between the selected percentiles.
Variance (ang. variance) \(-\) measures the degree of dispersion of the measurements around the arithmetic mean. It is calculated slightly differently for a sample and for an entire population (formulas in For those interested). Variance is always positive, but it is not expressed in the same units as the measurements.
Standard deviation (ang. standard deviation) \(-\) like variance, measures the degree of dispersion of the measurements around the arithmetic mean, but – unlike variance – is expressed in the same units as the data (it is the square root of the variance). The higher the value of the standard deviation (or the variance), the more diverse the group is with respect to the feature under study.

The coefficient of variation (ang. coefficient of variation)
The coefficient of variation, like the standard deviation, allows us to assess the degree of homogeneity of the group under study (formula in For those interested). It is a dimensionless quantity, which makes it possible to compare the variability of several groups with respect to the same feature, and of the same group with respect to several different features (expressed in different units). It is generally assumed that if the coefficient \(V\) does not exceed 10%, the features show statistically insignificant variability.
Standard errors (ang. standard errors) \(-\) these are not measures of dispersion of the measurement results, but rather indicate the degree of precision with which we can determine the values of population parameters from their estimators calculated from the sample. The most commonly used standard error is the standard error of the mean (ang. standard error of the mean, SEM) – formula in For those interested.

Other distribution attributes

Skewness, also called the coefficient of asymmetry (ang. skewness)
This measure indicates how much the distribution of the data differs from a symmetric distribution. The closer the value of the asymmetry coefficient is to zero, the more symmetrically the data are distributed around the mean. This coefficient usually lies within the range [-1, 1], although for particularly high asymmetry it may fall outside this range. Positive values indicate right-sided skewness (a longer right "tail"), while negative values indicate left-sided skewness (a longer left "tail"). The formula for skewness can be found in For those interested.

Kurtosis, also called the coefficient of concentration (ang. kurtosis)
This measure indicates how closely the spread of the data around the mean resembles the spread of data in a normal distribution. The higher the value of kurtosis above zero, the more sharply peaked the studied distribution is compared with the normal distribution; the lower the value below zero, the flatter the distribution is compared with the normal distribution. The formula for kurtosis can be found in For those interested.

Performing and interpreting the analysis

Descriptive summaries

When to use:

Descriptive summaries provide a quick way to prepare a report presenting a description of the data, ready to be inserted directly into a research paper. It is a tool that makes it easier to gather all the basic forms of data description in one place.

How to set up the analysis

Statistics \(\to\) Descriptive analyses \(\to\) Descriptive summaries.

Depending on how the measurements were made, one of three types of measure is most often used to describe a variable:

• mean \(\pm\) standard deviation, abbreviated \(\overline{x}\pm sd\)
• median [lower quartile; upper quartile], abbreviated \(Me [Q_1; Q_3]\), or median [min; max], abbreviated \(Me [min; max]\)
• count (percentage) of individual categories, abbreviated \(n (\%)\)

Depending on the needs and whether additional assumptions are met:

• on the interval scale, data can be described using any measure – researchers are also often interested in checking whether such data are normally distributed,
• on the ordinal scale, we can use medians (with quartiles or the smallest and largest value) or the counts and percentages of individual categories,
• on the nominal scale, only the counts and percentages of individual categories are available.

Performing and interpreting the analysis