Frequencies and descriptive statistics

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For each variable choose what to compute: Frequencies – a frequency table and a bar chart (for a categorical or ordinal variable); Measures – mean, standard deviation, median, range and skewness with a histogram (for a quantitative variable). The default is set according to the legend, and without it – according to the number of distinct values.

Guide

When to use?

Always – and it is always the first step. Pallant (2020, chapter 6) lists three tasks that descriptive statistics fulfill: describing the sample in the method section (how many participants, how many men and women and in what percentage, the age range and mean, education), checking the assumptions of the tests that will come later (mean, standard deviation, range, skewness), and answering research questions that are themselves descriptive ("What percentage of respondents are in favor?"). Gravetter and Wallnau (2017, p. 35) define its role simply: to organize pages of numbers into an understandable form, so that a pattern can be seen and communicated to others – and a frequency table is the most common way to do this. Field (2024, section 1.7) begins data analysis in the same place: count how many times each value occurs, draw the distribution, and only then summarize it in one or two numbers.

The page computes both types of description – a frequency table for a variable whose values are categories, and measures of central tendency and dispersion for a quantitative variable – and draws the appropriate chart for each. What is not here: a description of the relationship between two variables (correlations, scatter plot) and a crosstabulation of two categorical variables (the chi-square page). And a report of tables for all the survey questions at once – with a codebook and an intake report – is on the page Survey processing: codebook, checks and tables.

Which description suits which variable?

The decision derives from the level of measurement (the page Variables, levels of measurement and research designs). Pallant (2020, chapter 6) separates the two tracks in SPSS too: for a categorical variable – Frequencies, because "it makes no sense asking for the mean and standard deviation for gender or marital status"; for a continuous variable – Descriptives. Gravetter and Wallnau (2017, pp. 88-89) explain: on a nominal scale you can only count, so the mode (the largest category) is the only measure of central tendency; on an ordinal scale you can determine direction ("more" and "less") but not distance, so the median is suitable and the mean is not; and only on an interval or ratio scale, in which the distances between values are equal, do the mean and standard deviation say something.

Site convention for rating scales (the page Variables and levels of measurement): a single Likert item is an ordinal variable and is described in a frequency table; a multi-item score (the mean or sum of several items – the page Cronbach's alpha and scale score) is treated as interval and described with measures. The page's default works according to the legend ("nominal"/"ordinal" → frequencies, "scale" → measures), and without a legend – a numeric column with up to seven distinct values is considered a rating item; the decision can be changed for each variable with a click. A text column (for example "North / Center / South") is described with frequencies only – a mean of words does not exist.

How to read a frequency table?

A frequency table is the list of the variable's categories, and next to each one – how many participants are in it (Gravetter & Wallnau, 2017, pp. 35-36). The four columns of the table are exactly the four columns of Frequencies in SPSS (Pallant, 2020, Chapter 6):

Frequency (Frequency) – the number of observations in the category; the column total is N. Percent (Percent) – the frequency divided by all observations, including the missing ones, times 100; this is the "relative frequency" of Gravetter and Wallnau (p. 37). Valid percent (Valid Percent) – out of the valid observations only, and this is the percentage that is reported, because those who did not answer are not a "category"; when there are no missing values the two columns are identical, and when there are – the gap between them tells exactly how much is missing. Cumulative percent (Cumulative Percent) – the sum of the valid percentages up to this category; it is meaningful for an ordinal variable ("72.6% with education up to a bachelor's degree") and meaningless for a nominal variable, in which the order of the categories is arbitrary (Gravetter & Wallnau, 2017, p. 36: on a nominal scale the categories can be listed in any order). A "Missing" row and a "Total" row close the table as in SPSS, and the table note states on how many valid observations the valid percent is computed.

Pallant (2020, chapter 6) adds that attention should be paid to the size of the subgroups: very small or very unequal groups are a warning sign for the analyses that will come later (for example ANOVA by groups), and the frequency table is the place where this is seen first.

Measures of central tendency and dispersion

Mean (M) – the sum of the values divided by their number. It is the popular measure because it uses all the observations and is stable across samples (Field, 2024, Section 1.7.4), but it is affected by extreme values: in Field's example, one participant with 234 followers raised the mean by about 14 while the median moved only from 98 to 95.5. Median (Mdn) – the middle value after sorting; with an even number of observations – the mean of the two middle ones (Field, section 1.7.3; so too SPSS and this page). The median is hardly affected by extreme values and by an asymmetric distribution, and is also suitable for an ordinal scale. Gravetter and Wallnau (2017, pp. 86-88) add three situations in which the median, and not the mean, is the right measure: extreme values (income is the classic example), an undetermined value (a participant who did not finish the task – has no time, but does have a place in the order), and an open-ended category ("5 or more"). Mode – the most common value, the peak in the histogram; for a categorical variable it is the first category in the report sentence. Field (section 1.7.2) warns that there can be two or more (bimodal, multimodal) and that it is sensitive to a few single observations; Gravetter and Wallnau (pp. 88-89) point out that for a discrete variable it gives a value that actually exists ("a typical family with 2 children" and not "2.4 children").

Standard deviation (SD) – the typical distance of an observation from the mean: the square root of the mean of the squared deviations (Field, 2024, Section 1.7.5; Gravetter & Wallnau, 2017, p. 121). The page, like SPSS, divides the sum of squared deviations by n − 1 and not by n: a variance divided by n underestimates the population variance, and n − 1 corrects the bias (Gravetter & Wallnau, pp. 113, 117). A standard deviation that is small relative to the mean says the observations are close to it; large – that they are spread out; zero – that they are all identical (Field, Section 1.7.5). Range – maximum minus minimum; this definition, which SPSS also uses (Gravetter & Wallnau, p. 102), is very sensitive to a single extreme value, and so it complements the standard deviation and is not a substitute for it. What is not in the table: the quartiles and the interquartile range (IQR) – the range of the middle 50%, which is not affected by the extremes (Field, Section 1.7.5); Pallant (2020, chapter 6) asks to report it alongside the median when the distribution is asymmetric, and shows that it is obtained in Frequencies › Statistics › Quartiles.

Shape of the distribution: skewness

Before summarizing a distribution in a number, look at its shape (Field, 2024, Section 1.7.1; Gravetter & Wallnau, 2017, p. 48). A symmetric distribution can be folded in the middle; in an asymmetric distribution the observations pile up at one end and trail off in a tail at the other end. Positive skew – the tail is on the right, toward the high values (a difficult test, reaction times, income); negative – the tail is on the left (an easy test, satisfaction). Pallant (2020, chapter 6) gives examples from mental health: life satisfaction measures tend to be negatively skewed, and anxiety and depression measures in the general population – positively; this is a property of the phenomenon, not a flaw in the instrument.

The skewness measure in the table is G₁ – the SPSS formula, which gives 0 for a symmetric distribution, a positive value for a right tail and a negative one for a left tail; the farther the value is from zero, the more severe the skew (Field, 2024, Section 6.9.5). Next to it is its standard error (SE), also following SPSS. Field suggests a simple test: z = G₁/SE, and |z| > 1.96 is significant at the .05 level – and immediately qualifies it: in a large sample the test is significant even for negligible skew, and in a small sample it has no power (Box 6.7); therefore "in large samples don't do these tests – look at the shape of the distribution, interpret the skewness value, and thanks to the central limit theorem maybe don't worry about normality at all" (section 6.9.5). Tabachnick and Fidell (as cited in Pallant, 2020, Chapter 6) say the same thing: in a reasonable sample skewness "will not make a substantive difference in the analysis", and the way to judge is the histogram. The "Interpreting the result" box in the output gives z with this qualification.

A second sign of skew is the gap between the mean and the median: in a symmetric distribution they are almost equal; positive skew pulls the mean up but not the median, and if the mean is larger than the median – there is probably a right tail, and vice versa (Field, 2024, section 6.9.5; Gravetter & Wallnau, 2017, pp. 92-93: in a positively skewed distribution the order is mode → median → mean). Page convention: when |G₁| exceeds roughly 1 – the threshold at which Field (section 6.9.5) already speaks of "positive skew" in his hygiene example – the report sentence recommends also reporting the median, as Pallant asks for skewed data. The page does not compute kurtosis (the heaviness of the tails; Field, section 1.7.1, points out that it is often confused with the "peakedness" of the distribution); SPSS prints it next to the skewness, see the SPSS section.

Assumptions and warnings

No assumptions – a match between the measure and the scale. Descriptive statistics assume nothing about the distribution; the only question is whether the measure suits the level of measurement (section "Which description for which variable"). That is why the page warns when a variable marked in the legend as categorical is sent to "Measures" mode: a mean of nominal codes (1 = man, 2 = woman) is a number without interpretation (Gravetter & Wallnau, 2017, p. 88).

Missing values. Each variable is described on its own valid observations only – pairwise deletion, as the default of Frequencies and Descriptives; so N can vary from variable to variable, and the table says how much is missing. Pallant (2020, chapter 6) asks to check the percentage of missing values in each variable already at this stage and to ask why: much unexpected missingness in one variable, or a systematic pattern (women over a certain age who do not answer the age question), is a finding in itself. A non-numeric value in a numeric column (text, a dash, "999") is treated as missing and reported in a warning at loading with the row numbers; what to do with missing values – the page Missing values.

Extreme values. The range and the mean are sensitive to a single observation (Field, 2024, Section 1.7.4-1.7.5); Pallant (2020, chapter 6) recommends comparing the mean with the 5% trimmed mean of Explore, and checking whether the extreme value is a typo or a real observation. The page does not compute a trimmed mean or draw a box plot; the histogram shows single observations in the tail, and the page Outliers and assumptions explains what to do with them.

The histogram and the bins. A histogram suits a quantitative variable; for a categorical or ordinal variable a bar chart is drawn with gaps between the bars, because the categories are separate and their size cannot be assumed to be equal (Gravetter & Wallnau, 2017, pp. 43-45). The bin width in the histogram is set according to the rules of Gravetter and Wallnau (pp. 39-40) for a grouped frequency table: about ten bins, a "simple" width (1, 2, 5, 10...), each bin starting at a multiple of the width, and all of equal width – so the SPSS histogram, which chooses bins differently, will look somewhat different on the same data. The value (vertical) axis always starts at zero: cutting the axis exaggerates differences, and stretching it hides them (Gravetter & Wallnau, Box 2.1, p. 47; Field, 2024, section 5.2.1 – "show the data, don't distort them"). A histogram is not drawn for fewer than five observations or when all values are equal; a bar chart is not drawn for more than 25 categories – there the page warns that the variable may be quantitative.

What this page does not compute. The mode as a number (for a quantitative variable it is often meaningless, and SPSS prints the smallest of several), kurtosis, trimmed mean, confidence interval for the mean, quartiles and IQR, box plot, Q-Q plots and normality tests (Kolmogorov-Smirnov, Shapiro-Wilk) – all of them are provided by SPSS Explore (Pallant, 2020, Chapter 6; Field, 2024, Section 6.9), and on the normality tests see the qualification in the previous section. Description by groups (the mean age for men and for women separately) – on the t-test and ANOVA pages.

And how do you do it in SPSS?

A note on versions. The procedures in this section are stable across versions, but the location of menu items and the names of options may change slightly. Pallant (2020) was written for version 26 and Field (2024) for version 29; the three procedures – Frequencies, Descriptives and Explore – sit in the same menu in all versions.

Categorical variable (Pallant, 2020, Chapter 6): Analyze › Descriptive Statistics › Frequencies… – move the variables to the Variables box, and OK. With the Charts… button you can ask for a bar chart (Bar charts, in percentages). The syntax:

FREQUENCIES VARIABLES=sex education
  /BARCHART PERCENT
  /ORDER=ANALYSIS.

Quantitative variable (Pallant, Chapter 6): Analyze › Descriptive Statistics › Descriptives… – the variables to the box, and with the Options… button check Mean, Std. deviation, Minimum, Maximum, Range, Skewness (and Kurtosis if you want). Descriptives does not print a median; it is obtained in Frequencies › Statistics… › Median (and there also Quartiles), with Charts… › Histograms and without the frequency table (Display frequency tables unchecked). This is exactly how the syntax the page produces is built:

DESCRIPTIVES VARIABLES=age
  /STATISTICS=MEAN STDDEV MIN MAX RANGE SKEWNESS.
FREQUENCIES VARIABLES=age
  /FORMAT=NOTABLE
  /STATISTICS=MEDIAN
  /HISTOGRAM.

Reading the output. In the Frequencies table: Frequency, Percent, Valid Percent, Cumulative Percent are the four columns of the table on the page, and the Missing System row is the "Missing" row. In the Descriptive Statistics table: N = valid N, Minimum, Maximum, Range, Mean, Std. Deviation as their names say, and Skewness in two columns – Statistic and Std. Error – which are the skewness and the standard error in the measures table. Pallant (chapter 6) points out that in reporting the values are rounded to two decimal places. The Valid N (listwise) row at the bottom of the table is the number of participants who have a value in all the variables together (listwise deletion – Field, 2024, section 6.9.2) – not the N of any single variable.

More in Explore (Pallant, Chapter 6; Field, 2024, Section 6.9.2): Analyze › Descriptive Statistics › Explore… gives in one table also a 95% confidence interval for the mean, the 5% trimmed mean, variance, interquartile range and kurtosis, and with the Plots… button – a histogram, a box plot and a Q-Q plot with the normality tests. Field reminds you to select Exclude cases pairwise in Options, otherwise Explore will use only participants who have a value in all the selected variables. The "Download SPSS syntax" button in the output produces all the steps – loading, dictionary (if there was a legend), FREQUENCIES and DESCRIPTIVES – on the loaded data and with the loaded variable names.

And how do you do it in R?

In base R: table() for frequencies and prop.table() for percentages (missing values are omitted from table by default, so the percentage obtained is the valid percent), mean(), sd() (divides by n − 1, as in SPSS), median(), range(); skewness in the SPSS formula is skewness(x, type = 2) from the e1071 package. The script was run verbatim on the sample file and gave the table values (in the comments). The site rule for R: the file is saved in UTF-8, every comment line starts with an English word, and Hebrew sheet and variable names (as in the Hebrew sample file used here) require a UTF-8 environment (on Windows: R 4.2 or later, or Sys.setlocale('LC_ALL', 'Hebrew')).

# Frequencies and descriptive statistics – Professor Stat. This file is saved in UTF-8; every comment line starts with an English word
library(readxl)   # reading Excel
library(e1071)    # skewness in the SPSS formula: skewness(type = 2)

# Loading: the standard template – row = participant, column = variable; the "data" (data) sheet
d <- read_excel("C:/Rdata/DATA_freq_example_eng.xlsx", sheet = "data")

# Frequency table for a categorical variable: frequency, percent (of all, including missing), valid percent and cumulative percent
f <- table(d[["education"]])
cbind(f, percent = round(100 * f / nrow(d), 1), valid_percent = round(100 * prop.table(f), 1), cumulative = round(100 * cumsum(prop.table(f)), 1))
# Result: 17 (14.2, 14.5, 14.5) · 18 (15.0, 15.4, 29.9) · 50 (41.7, 42.7, 72.6) · 32 (26.7, 27.4, 100.0)
sum(is.na(d[["education"]]))   # missing = 3

# Descriptive measures for a quantitative variable – on the valid observations only
x <- na.omit(d[["age"]])
c(n = length(x), mean = mean(x), sd = sd(x), median = median(x), min = min(x), max = max(x))
# Result: 116, 39.40, 12.97, 39.5, 18, 75
skewness(x, type = 2)                                   # skewness G1 as in SPSS = 0.213
n <- length(x); sqrt(6 * n * (n - 1) / ((n - 2) * (n + 1) * (n + 3)))   # its standard error = 0.225

# Histogram with the same bins as on the page: width 10, starting at a multiple of 10, bin closed on the left [10,20) …
hist(x, breaks = seq(10, 80, by = 10), right = FALSE)$counts   # 11 19 28 35 17 4 2

Reporting the results

In what form. Field (2024, section 1.8.3) gives the APA rule of thumb: up to three numbers – in a sentence; four to twenty – in a table; more than twenty – in a figure; and do not repeat the same numbers in two forms. So the description of the sample (gender, education, age) usually goes into one table in the method section, on the model of Table 7.2 in the APA manual (American Psychological Association [APA], 2020, Section 7.21): the variable name as a heading row, the categories indented below it, and for each category an n column and a % column; in a column headed "%" the numbers are written without the percent sign (section 7.12), and the overall N is given in the title or in the table note. The tables in the output follow the APA table rules: the table number in bold and its title in italics (section 7.11), horizontal lines only (section 7.17), and a general note below (section 7.14).

In a sentence. A categorical variable: "Most participants (n = 68, 56.7%) were women"; a quantitative variable: mean and standard deviation together – M = 39.40, SD = 12.97 – because a mean without a measure of dispersion is half the information (Field, 2024, Section 1.8.3; Gravetter & Wallnau, 2017, pp. 89, 121); for skewed data – the median with the range or IQR: Mdn = 36, IQR = 21 (Field, Section 1.8.3; Pallant, 2020, Chapter 6). The symbols M, SD, Mdn, N, n need no definition (APA, 2020, Section 6.44); N is the whole sample and n a subgroup. The page prints means and standard deviations to two decimal places, as SPSS does; APA (section 6.36) adds that for data on an integer scale – such as questionnaire items – one decimal is enough, and that percentages and measures of central tendency are rounded "as much as possible" without losing precision. Skewness is reported when it is relevant to the assumptions: skewness = 0.21, SE = 0.22, with a zero before the decimal point because the measure can exceed 1 (section 6.36).

In a figure. A chart is a figure according to APA: a number, an italic title, the variable names on the axes, and a figure note stating what each symbol represents and what N is (APA, 2020, Sections 7.22-7.28) – this is how the charts in the output are built, and this is how they paste into Word (the note below the chart). Field (2024, section 5.2.1) quotes Tufte: show the data, don't distort them, without "chartjunk" – no three-dimensional effects, shadows or backgrounds.

Common mistakes

A mean for a nominal variable. "The mean gender is 1.57" – the codes are names, not quantities (Gravetter & Wallnau, 2017, p. 88; Pallant, 2020, Chapter 6). Report frequencies and percentages.

Reporting percent instead of valid percent. When there are missing values, "percent" is computed from all participants including those who did not answer; the percentage that describes the respondents is the valid percent – and the number of missing values is reported separately.

A mean without a standard deviation, a median without a range. A measure of central tendency alone does not describe a distribution (Field, 2024, Section 1.8.3): two lecturers with the same mean of 2.6 and standard deviations of 0.55 and 1.82 are two different stories (Field, Section 1.7.5).

A mean for a highly skewed distribution. Income, reaction times, number of children – the mean is pulled toward the tail; report the median (Gravetter & Wallnau, 2017, pp. 86-87; Pallant, 2020, Chapter 6), and if you want – the mean as well, noting the skewness.

Interpreting a single Likert item as an interval variable. Site convention: a single item is ordinal – a frequency table and the median; a multi-item score – mean and standard deviation (the page Variables and levels of measurement).

Testing normality with a significance test in a large sample. The test will be significant even for a negligible departure (Field, 2024, Box 6.7; Tabachnick & Fidell, as cited in Pallant, 2020, Chapter 6). Look at the histogram and the skewness value.

A value axis that does not start at zero, a three-dimensional chart. Both distort what the reader sees (Gravetter & Wallnau, 2017, Box 2.1, p. 47; Field, 2024, Section 5.2.1).

Describing the sample from Valid N (listwise). This is the number of participants complete on all the variables together – listwise deletion (Field, 2024, Section 6.9.2); the N of each variable is the one in its own row.

References

American Psychological Association. (2020). Publication manual of the American Psychological Association (7th ed.). https://doi.org/10.1037/0000165-000

Field, A. (2024). Discovering statistics using IBM SPSS statistics (6th ed.). SAGE.

Gravetter, F. J., & Wallnau, L. B. (2017). Statistics for the behavioral sciences (10th ed.). Cengage Learning.

Pallant, J. (2020). SPSS survival manual: A step by step guide to data analysis using IBM SPSS (7th ed.). Routledge.

Tufte (2001) – as cited in Field (2024); Tabachnick and Fidell (2013) – as cited in Pallant (2020). According to the site rule, the reference list includes only sources that were read in the original. Field (2024) and Pallant (2020) in a reflowable digital copy – cited by section and chapter; Gravetter and Wallnau (2017) and APA (2020) – by page and section of the printed edition.