## Statistics

The Results section is arguably the most important part of a research report.  When a professional scientist reads a research report, the first section that he or she typically goes to is the Results.  Likewise, when a company or government agency has commissioned a research project, they are most interested in the results obtained. In the Results section, the major findings of the study are described through the use of descriptive statistics and, if relevant, with inferential statistics. Visual aids such as tables and/or figures are often included to allow for efficient communication of the outcomes.

##### What you may need to include in the Results
Relevant information that can accompany the presentation of the results can include:

• What data screening procedures were carried out (e.g., data entry checks, checking of the necessary conditions of the analyses through the use of boxplots, histograms, etc.) and the outcomes of these procedures.

• What descriptive statistics are reported including a justification for the choice.

• What inferential statistics were calculated.  This information will be closely linked to the design of your study and what hypotheses were tested.  For instance, an observational (or correlational) research approach will typically analyse the results using correlational methods.

• The probability criterion for any tests of significance (i.e., the probability level).

The Results section will summarise the data collected (descriptive statistics) and the statistical analyses on the data that seek to examine the relationships among the variables and relate these to the population of interest (inferential statistics).  Basic principles in presenting the results are given below. Subsequent sections focus on more specific aspects of the Results.

• Give enough detail to justify your conclusions.

• Present all the relevant results, even when these do not support your hypotheses or are not statistically significant.

• Present the results either within the text, in tables, or in figures.  Choose the medium that is most appropriate and will facilitate communication. For instance, for reporting of uncomplicated descriptive statistics you might present them in the text.  Tables can be used for more complex statistics. Figures are excellent to highlight similarities, differences, or patters in a data set.

• Avoid repeating the same results in different places.  For instance, do not present the same descriptive statistics in a figure and in the main text. You could report complementary statistics, however (e.g., one figure showing a histogram and a description of the mean and standard deviation in the text)

• If you present descriptive statistics, tell the reader what to look for.  For instance, describe the figure in terms of the differences and similarities, using terminology consistent with that used in the figure.

##### Preliminary: Using Numbers
The Results section will include the reporting of numbers. Here are a few guidelines to keep in mind when using numbers.

1. Although there are some exceptions, you generally use digits to express numbers 10 and above and use words to express numbers less than 10.  For example, “14 cm, 23 books, nine individuals, two words”, but note the next point.

2. You also use digits to express

• All statistical copy, for example r = .34

• All numbers below 10 that are grouped for comparison with numbers above 10, for example “2 of the 14 measures of skewness were….”

• All numbers that precede a unit of measurement, for example 14 cm

• All numbers that represent statistical or mathematical functions, for example “the 1st quartile”

• All numbers that represent times, dates, ages, sample sizes

• Numbers that describe a figure or table in the text, for example “Figure 1”, “Table 1”.

3. Use words to express numbers that begin a sentence.  For example “Twenty books were selected….”.

4. Generally, use a zero before the decimal point when numbers are less than 1, for example 0.24 cm, but do not use a zero before a decimal point in cases when the number cannot be greater than 1, for example correlations, proportions, statistical significance, such as p < .05.

5. In general report to two decimal places.  You may use less or more than two decimal places when this is appropriate for facilitating communication (e.g., using less than two decimal places) or maintaining statistical precision (e.g., using more than two decimal places)

View the video below for how to use numbers in text.

##### Statistical Copy
1. When reporting inferential statistics, give enough information so that the reader can fully understand the analyses conducted, their interpretation, and possible alternative explanations for the results.

2. If reporting inferential statistics based on a comparison (e.g., t statistic), report the value of the test statistic, degrees of freedom, the probability of obtaining a value as extreme or more extreme than the one obtained, and the direction of the effect (e.g., what mean is larger than what other mean?).

3. You may have seen two different approaches to reporting statistical significance.  They differ in what method is used to report the probability level associated with the test statistic. For the outcomes of a significance test, you might either report the exact probability level obtained or whether the probability was less than the predetermined alpha level.  APA guidelines indicate that, in general, the exact probability should be stated, e.g., F(1, 1098) = 5.85, p = .016.  When the probability is very small, you can also report that the significance level is less than a certain value, e.g., F(1, 1098) = 14.28, p < .001.
##### Statistical Symbols

1. Use Greek letters to denote population parameters (e.g., μ) and italicised Latin letters to denote sample statistics (e.g., M).

2. Use an uppercase, italicised N to represent the total sample size, and a lowercase, italicised n to represent the number in a portion of the total sample (e.g., the sample size of a group).

3. Except for Greek letters and vectors, present statistical symbols in italics e.g., N, df, p, SD, F, t.

4. Use common abbreviations for statistical terms and symbols e.g.,

• ANOVA = analysis of variance

• M = mean

• SD = standard deviation

• SE = standard error

• p = probability

• t = computed value of a t statistic

• Mdn = median

• Q = quartile

• df = degrees of freedom

5. Space statistical copy as you would with words.  In general use spacing between numbers and other text, e.g., F (1, 23) = 3.53, p < .05.

View the video below on how to insert statistical symbols in Microsoft Word.

##### Tables
Tables are very efficient because they allow the author to present a large amount of data in a very small space.  An advantage of tables over figures is that they can show exact numerical values, although it can sometimes be difficult to determine the relationship between values.  Tables typically allow the read to examine the relationship between values better than when these are presented in the main body of the text.

1. You should always refer to your table in the text e.g., “As shown in Table 1, the mean reading ease….”.

2. Number the tables in the order in which they are presented in the text e.g., “Table 1”.

3. The table should have a title.  This title is place above the table and it should provide a brief but clear description of the contents in the table.

4. Use headings and subheadings to clarify the identity of items in a table.  Every column and should have a heading.  Some rows will also require a heading.

5. Use only horizontal lines to separate elements in a table.  Do not use vertical lines.

6. When reporting variability, you typically report the standard deviation rather than the standard error or variance.

• It is good practice to include the variable names in the table titles. Avoid abbreviations of variable names.

• Correlation tables only need to include the correlation values in the upper diagonal since the values for the lower diagonal will be identical. The values for the diagonal are listed as “-”

• If the sample size is identical for each correlation, do not list the sample sizes within the table.  Rather, give the sample size in the table title.

• You can use an “*” and “**” to symbolise whether a correlation is significant at the .05 or .01 level, respectively. The exact probability values can be given in the text when you describe the correlations.

• The units for each measure should be given in the title or in the table itself.  It is redundant to give them in both.

• If you cannot get your correlation table to fit on the page, try putting it on a page that is landscape orientated.

• It is good practice to put your most important variable (usually the response variable) in the first column and row of a correlation table

For more on tables, see the instructions for creating and formatting tables.
##### Figures
Figures are an excellent way to show large amounts of data and to show the relationships between variables. However, figures typically do not typically give exact numerical values and are more difficult and time consuming to prepare than are tables.

1. A figure should have a title (the same as a table).  This is positioned above the figure and it should provide a brief but clear description of the contents of the figure.

2. The figure should be legible, and the labels and numbers should be large enough to be easily read.  Typically, the font size will be 10 to 12 point font in a consistent font with the body of your text.

3. Label the axes and key (if relevant).

4. Bar graphs are generally used when the independent variable is categorical.  Line graphs are generally used to show the relation between two quantitative variables or changes over time.

5. The axes should be proportioned appropriately, such that the x-axis is longer than the y-axis.

6. Use a good software package to produce the figure e.g., SPSS, MS Excel.

• A figure has a title

• Figures should be numbered consecutively.  Likewise, tables should be numbered consecutively.  Use numerals 1, 2, 3, ….etc. not i, ii, iii, iv….. or A, B, C, D…… or a, b, c, d……

• It is good practice to include the variable names in the figure title. Avoid using abbreviations.

• The units for each measure should be given in the title, notes, or in the figure itself.  It is redundant to give them in both.

• If you have the same variable on several figures, it is good practice to use the same scale for that variable across all figures.  This helps to compare one figure to another.

• Figures should be large enough to be easily read.  This is particularly important for the size of the axis labels.  Use SPSS to change the figure before you cut-and-paste it into your word processing document.

• If you are showing the mean values, always show a measure of variability. This should either be the 95 confidence interval of the mean or the standard error of the mean.

• Resist the temptation to get too fancy! The best figures are the ones that communicate clearly. Not the ones that have the prettiest colours!