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Quantitative Data

16 Even the most basic social science data can be expressed numerically and tested statistically. This understanding sees many areas of qualitative research that historically contained very little quantitative data or analysis (anthropological case studies, for example) now, sometimes, using statistical tests.

A strength of quantitative research is that detailed rules encourage care. The rules get very complicated, but every statistical test has procedures that others can replicate. This provides an intellectual discipline that encourages accuracy. If we pay careful attention to the procedures and rules, the work will be systematic and thorough. Many researchers, including myself, are without a strong mathematical background and find that statistics are difficult. This is not a reason to avoid them; it is a challenge. There is a steep learning curve, but you can become quite proficient if motivated. The first data chapter in my first major research project took two months to write because I had to teach myself statistics; but the last one only took two days.

Care with the maths does not necessarily make the research strong. Overdoing sophisticated statistics to make minor studies appear important is merely statistical overkill. A statistically significant relationship between an independent and a dependent variable is only as useful as the underlying analysis. If we are intellectually sloppy and overlook valid alternative variables, our research is of little value despite the statistics. In other words, statistical analysis can add to the reliability of our research, but we need to establish validity too: we need analytical rigour as well as technical strength.

This chapter will:

1. review the key principles of quantitative data established earlier in this book; and 2. outline some basic techniques for manual and computer analysis of descriptive

and inferential statistics.

With a few exceptions that demonstrate some key principles or help with spreadsheets, the chapter will not go far into the definitions or mathematics. If you are going to do

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EBSCO Publishing : eBook Collection (EBSCOhost) - printed on 1/22/2022 2:28 AM via PURDUE UNIVERSITY GLOBAL AN: 340339 ; Gerard Guthrie.; Basic Research Methods : An Entry to Social Science Research Account: ns019078

168 Basic ReseaRch Methods

some quantitative analysis, find a textbook on statistics that is appropriate for your level, and make good use of the internet.

16.1 Quantitative data principles

Like qualitative data, measurement principles will often help solve problems that arise during analysis. Does a table seem too complicated? Not sure where some material belongs? Just like words, the chances are that an answer lies in confusion over multiple variables with different measurement properties or underlaid by inconsistent semantic differentials. The underlying principles that apply to numbers and tables are:

1. Numerical data expresses quantities and variables (Chapter 14). 2. Numbers come from some types of available data (Chapter 9), structured obser-

vation (Chapter 10), questionnaires (Chapter 12) and tests (Chapter 13). 3. Numbers can be classified on all measurement scales, but in social science we

mainly use the nominal and ordinal scales (Table 14.1). 4. The further up the scales, the more mathematical information is added, the

more precise the measurement and the more powerful the statistical tests that can be used to test null hypotheses (Chapter 14).

The basic numerical steps to be followed for quantitative data are the same as for qualitative data. First, describe; then, analyse; and later, draw conclusions. Two types of statistics do the description and analysis:

1. Descriptive statistics, such as percentages and means, summarise the numbers and can be represented in graphs.

2. Inferential tests analyse statistical significance for testing hypotheses and drawing inferences about the strength of the findings.

An inferential test with a probability (p) greater than .05 (for example, .01) shows a significant difference. This is useful when we are predicting differences. However, if we are not looking for a difference, a result between p = .99 and p = .06, is the desired outcome, for example, because it shows the sample is not significantly different from the population from which it was drawn.

One technical issue is the choice of test types. Inferential tests further divide into two types. Parametric tests are based on an assumption of a normal distribution in the data and, technically, they are based on the mathematical properties of interval or ratio data. Non-parametric tests do not make an assumption of normalcy and, thus,

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are especially useful with small samples that are not normally distributed and with lower level data. Non-parametric statisticians have argued that tests designed for data at the higher interval and ratio levels should not be used with the ‘weak measurement’ provided by the lower nominal and ordinal levels typically found with social science data. Subsequently, the proponents of ‘strong statistics’ demonstrated that the mathematical assumptions of many common parametric statistics are strong enough to allow their use to extract more information on statistical significance from nominal and ordinal data than is available from non-parametric tests.

The effect is that many statistical tests based on higher measurement scales can use data from lower scales. Nonetheless, the results should be interpreted according to the underlying scale. For example, ordinal data remains ordinal even if tested with a measure originally designed for interval data. Just because the test gives a significant result does not mean that the ordinal data now shows exact intervals. It still continues to show results that are greater than or less than, but not by any particular amount.

Additionally, underlying the use of inferential statistics is an argument that goes back to the methodological issues in Chapter 4. Post-positivist researchers object that parametric statistics reflect the law-seeking normative assumptions of positivist research, and that this is contrary to the effort in naturalistic research to emphasise the uniqueness of participants. They will often admit non-parametric statistics, which do not assume normalcy and are useable with small non-normal samples.

The first step in choosing which statistics to use is to identify the data’s measurement scale. Table 16.1 shows some descriptive and inferential statistics that can be used with the key measurement scales. For example, binary data should use the mode (the most common score) as the measure of central tendency, which can be illustrated with column graphs. The binomial inferential test can be used to test significance. In fact, there are scores of statistical tests for all five measurement scales, but this book only identifies a few basic ones that are commonly used with nominal and ordinal data and are acceptable for much basic research. The table shows non-parametric (NP) statistics as well as common parametric (P) ones, which can be used with the lower data levels.

For computer analysis of numbers, there is the same choice as for words: use whichever package is available on your system, buy a programme and learn how to use it, or use a spreadsheet. Common statistics packages include BMDP, SAS and SPSS, but they are complex. Spreadsheets are a practical, but sometimes clunky alternative. I will use examples from Excel 2003, which has extensive capabilities for both descrip- tive and inferential calculations. To use them, make sure that the statistic functions are activated at ‘Tools’ > ‘Add-Ins’ > ‘Analysis Tool Pak’. This will activate a ‘Data Analysis’ command in the ‘Tools’ menu, which will give some of the statistical tests available. You can also use the Excel search function and search ‘statistical analysis’, ‘function’ and ‘formula’ to get overviews, and make use of the ‘Help’ function to search particular

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measures such as ‘mean’ or ‘chi square’. ‘Insert’ > ‘Function’ > ‘Statistical’ also lists the available functions.

The first step is to input the data into spreadsheet cells. Beforehand, set up a check of data entry mistakes at ‘Data’ > ‘Validation’, which will alert you to any entries outside the data range, but check entries anyway as this tool will not alert you to mistakes within the permissible range. With a questionnaire, column headings should be the question numbers that contain numerical responses. Row names should be the questionnaire code numbers. Perform statistical functions on columns (which now contain all the answers to particular questions) at cells underneath each one.

Table 16.1 Basic statistical measures

Scale Descriptive statistics Function

Binary 1. Mode (NP) 2. Column graphs

1. Most frequent score 2. Visual comparison

Nominal 1. Median (NP) 2. Mean (P) 3. Standard error of the mean (P) 4. Column graphs 5. Line graphs 6. Pie charts

1. Centremost score 2. Arithmetic average 3. Statistical error in the sample mean 4. Comparisons 5. Trends 6. Proportions

Ordinal 1. Mean (P) 2. Standard error of the mean (P) 3. Histograms 4. Pie charts

1. Arithmetic average 2. Statistical error in the sample mean 3. Comparisons 4. Proportions

Inferential statistics Function Binary Binomial test (NP) Sample vs population proportions

Nominal 1. Chi square (NP) 2. Contingency coefficient (NP) 3. t test (P) 4. F test (analysis of variance or ANOVA) (P) 5. z test (P)

1. Observed vs expected frequency 2. Correlation between two variables 3. Two sample means 4. Three-plus sample means 5. Sample vs population means

Ordinal 1. Chi square (NP) 2. Spearman rank correlation coefficient (NP) 3. Kendall coefficient of concordance (NP) 4. t test (P) 5. F test (ANOVA) (P) 6. z test (P)

1. Observed vs expected frequency 2. Correlation between two rankings 3. Correlation among three-plus rankings 4. Two sample means 5. Three-plus sample means 6. Sample vs population means

Source: Author. Note: NP: non-parametric; P: parametric.

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16.2 Descriptive statistics, tables and charts

Now you are ready to find out what the data says. The following list has several ways of using descriptive statistics to summarise data. Each of these adds more information, so check the data against each one:

1. Central tendency or ‘average’ (mean, median or mode). 2. Distribution or indicators of the spread of the data (standard deviation, quartile

deviation). 3. Outliers or extremes (the topmost and bottommost scores). 4. Range (the difference between the top and bottom scores). 5. Non-conforming cases (data that appear not to fit the pattern).

Table 16.2 has Excel functions for common descriptive statistics.

Table 16.2 Guidance on descriptive statistics

Function Purpose

AVERAGE Arithmetic mean CORRELATION Correlation between two sets of data COUNTIF Tallies cells with data meeting particular requirements FREQUENCY Tallies cells containing particular numbers MEDIAN Middlemost number MODE Most frequent number PERCENTILE Percentile values PERCENTRANK Percentile ranks QUARTILE Quartile deviation RANK Ranks STDEV Standard deviation SUM Totals ‘Data’ > ‘Filter’ Identify particular values ‘Data’ > ‘Sort’ Order data alphabetically or numerically

Source: Author.

Next, use appropriate summary statistics to set up tables in the word processor and then use the table data to create graphs. Tables and charts are equivalent to paragraphs. Just as a paragraph deals with one main idea, so does a table or a chart by presenting data about one particular aspect of the research. Usually, a table will present numbers set out in rows and columns. A chart or figure will present ideas in a systematic form such as a diagram. Normally, each of these will require one or two

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written paragraphs of explanation. The following list will help improve the quality of tables and charts:

1. Avoid too much information in each one; if necessary, split data or ideas into smaller units and present them in two or three tables.

2. Identify each table, chart or figure.

(a) Number each one. (b) Use a clear and accurate title.

3. Label each row and column accurately, and show the units used in each row and column in the table (for example, per cent or No.).

4. Space out tables and figures—avoid clutter. 5. Keep tables on one page.

(a) Do not split tables, figures or charts over more than one page. If they are longer than a full page, divide the data into two tables.

(b) Start tables straight after the end of the paragraph that first refers to them if they fit on that page. If not, place them after the paragraph that carries onto the next page.

6. Cite sources: cite the sources for your information in a table note, including your own research.

Table 16.3 demonstrates these points.

Table 16.3 Reporting of most troublesome incident to the police

Reported Arawa

2004 (%) Arawa

2005 (%) Arawa

2006 (%) Buka

2004 (%) Buka

2005 (%) Buka

2006 (%)

Yes 16 15 35 23 18 19 No 84 85 65 77 82 81 Total 100 100 100 100 100 100

Source: Guthrie et al. (2007: 55). Note: Q.4.13. Arawa 2004 N = 106, Non-response = 66%; 2005 N = 106, Non-response = 65%;

2006 N = 75, Non-response = 75%. Buka 2004 N = 125, Non-response = 57%; 2005 N = 94, Non-response = 68%; 2006 N = 74, Non-response = 75%. The high non- response rates derive mainly from respondents who gave nil responses to S.3.

The safest Word graph types are:

1. Clustered column (to compare scores). 2. Line (for trends). 3. Pie charts (for proportions).

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For graphs, take time and explore fully the options for each type, setting up the first graph very carefully. Thereafter, you will have the style you want, into which you can cut, paste and input new data. This way other such graphs will be much easier to prepare. You can use Excel to set up the graphs and insert them from Word using ‘Edit’ > ‘Paste Special’. Or, you can use the Word graph functions, which are quite advanced. With Word, copy the table data into the datasheet, accept the graph provided, then right click on it to change and/or edit it with ‘Chart Object’ > ‘Chart Type’ > ‘Edit’. Then, right click the graph again and use ‘Chart Options’.

Use distinct colours if your printer supports them, but it is easy to be carried away with other options. Some graph types are better for public relations than for research. In particular:

1. Do not use three-dimensional graphs. They use depth, which implies volume and distorts the visual perception compared to, say, a bar graph.

2. Set up all vertical axes so that they start at zero and do not distort rates of change.

3. Use logarithmic line graphs to show rates of change.

16.3 Inferential statistics

Inferential statistics allow inferences to be drawn about the similarities or differences between the sample and the population, or between samples or between subsets of a sample. Testing can use:

1. Means (for example, t and z tests). 2. Variance (for example, analysis of variance or ANOVA). 3. Distribution (for example, chi square). 4. Correlations (for example, Spearman rank correlation coefficient).

An important point to note is that a correlation measures a relationship between two variables, usually on a scale that ranges from +1.00 to –1.00. A high positive correlation means the variables change in the same direction; a negative correlation means that they change in opposite directions. Some correlation coefficients can be tested for significance and correlations can be used for prediction (if Variable A changes, so too does Variable B). However, this describes an association between the variables and does not establish cause-and-effect unless measured as part of an experimental design. The changes could be coincidental. For example, a significant positive correlation might be found between smoking and alcoholism, but smoking does not cause alcoholism. Generally, negative correlations occur between health and education, on the one hand,

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174 Basic ReseaRch Methods

and financial poverty on the other. But lack of health and lack of education do not cause financial poverty; lack of money does.

Key issues to be considered while choosing a significance test or a particular version of a test are:

1. The measurement level (binary, nominal, ordinal, interval, ratio). 2. Number of sample cases (one sample, two sample, or k (three-plus) samples). 3. Sample type (related or independent). 4. Sample size. 5. Direction of hypothesised difference (one-tail or two-tail).

Statistical tests can be extremely complicated and require algebra to be understood. However, many of the basic statistical tests are quite straightforward and can be calculated with Excel (Table 16.4). Study carefully the ‘Help’ material on each function because it is written in mathematic language. You will also have to calculate and enter degrees of freedom to get test results (I would like to be able to explain what they are, but like many, I have failed to understand the numerous definitions. Have faith and just do what the textbooks say!).

Table 16.4 Guidance on inferential statistics

Function Purpose

CHIDIST Chi square statistic CHITEST Chi square significance CORREL Correlation coefficient CRITBINOMIAL Binomial test FDIST F test statistic FTEST F test significance NORMSDIST z score TDIST Student’s t statistic TTEST Student’s t test significance ZTEST z test significance

Source: Author.

16.4 Presenting data

The steps in quantitative data presentation are similar to those used for qualitative data:

1. Describe the numerical results (with descriptive statistics in the data chapters). 2. Analyse (with inferential statistics in the data chapters). 3. Interpret (with written interpretations later in the conclusions chapter).

Clarity and brevity are important, as is neutrally toned language. C op yr

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Box 16.1 is an example combining many of the features discussed in the previous section. This is a synthesis of results from 16 crime surveys on one particular indicator for levels of reported property crime.

1. The first sentence in the box describes the indicator by defining the terms. 2. The second sentence states the importance of the indicator. 3. The first dot point is a brief analysis using statistical significance. Chi square

was used to test whether or not there was a common national level of the crime, which the result showed was not the case; the point being that normalcy was tested, not assumed.

4. The second paragraph briefly summarises the data shown in the graph, identifying the top, middlemost and lowest ranking towns to express both the average and extreme cases, and then states the national mean. The third sentence comments on exceptions and patterns.

5. The graph presents the data. The heavy line shows the national mean, with the columns showing the towns relative to the mean and to each other. The graph was edited to insert the raw percentages at the top of each column so that a separate table was unnecessary.

6. The paragraph following the graph discusses an issue that arose, and then briefly expresses what it meant to the respondents by quoting some of their comments, thus adding qualitative data to the quantitative. With other indicators, tables were used to present the data for the top, middlemost and lowest ranking towns to demonstrate the amount of variation.

16.5 Summary

The most basic social science data can be expressed numerically and tested statistically. A strength of quantitative research is that its detailed rules encourage intellectual discipline.

Quantitative data principles

1. Numerical data expresses quantities and variables. 2. Numbers particularly come from some types of available data, structured

observation, questionnaires and tests. 3. In social science research, quantities are usually classified on the nominal and

ordinal measurement scales. 4. Inferential statistics are used to test null hypotheses. Parametric tests are based

on an assumption of a normal distribution in the data. Non-parametric tests do not make an assumption of normalcy.C

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176 Basic ReseaRch Methods

Box 16.1 ComBining types of analysis

Levels of Property Crime

‘The…indicator for particular types of crime victimisation was the mean percentage of households that had a member who was a victim of stealing property (Graph). In all the surveys, theft was the most common type of crime victimisation reported.

• The differences from the national mean were highly significant (X2 = 111.5, df = 15, p > .001), ie. the towns had very different levels of victimisation involving theft.

‘The highest rate was in Kainantu, with two-thirds (67%) of households in 2008 being victims. Port Moresby in 2004 was the middlemost town, with 38%. Arawa in 2006 had 8%. The national mean was 38.3% of households being victims in the previous 12 months. Notable was the relatively high rate for Kokopo, and the declines in Arawa and Buka from 2004 to 2006, which were statistically significant (Arawa X2 = 17.8, df = 1, p = .001; Buka X2 = 12.5, df = 1, p > .001).

Graph: Household the victim of stealing

‘During the surveys, informal comments were made occasionally to the researchers that petty theft was not a “real” crime: either it was a traditionally derived behaviour because private ownership was not a feature of tribal life; or it was so common as to be part of daily life and not really a crime. However in all the surveys, comments in the interviews about the most troubling crimes that had occurred to respondents in the previous year showed that stealing was a constant irritation... For example:

• I paid a lot for the bicycle. • We paid a lot of money for that generator. • Those bags of dry beans were worth K1000… • When I recovered the property it was damaged. • They stole our clothes on the line. • Truck parts are really expensive today. People lack respect for others’ belongings.’

Source: Adapted from Guthrie (2008: 30–31).

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5. The mathematical assumptions of many common parametric statistics are robust but the results should still be interpreted according to the underlying measurement scale.

Descriptive statistics, tables and charts

1. Summarise data using central tendency, distribution, outliers, range and non- conforming cases, as appropriate.

2. Tables and charts present data about one particular aspect of the research. Simple column, line and pie charts are the most useful graphs.

3. You should: (a) not put too much information into each table, chart or figure; (b) identify each one; (c) accurately label each row and column; (d) space out tables and figures; (e) keep tables on one page; and (f) cite sources.

Inferential statistics

Key issues in choosing a test or a particular version of a test are the measurement level, number of sample cases, sample type, sample size and direction of hypothesised difference.

Presenting data

Describe the numerical results with descriptive statistics, then analyse with inferential statistics, then make written interpretation. Clarity and brevity are important, as is a neutrally toned presentation.

To repeat a key point, just because we are careful with our math does not necessarily mean that our research is strong. If we are intellectually sloppy and overlook valid alternative variables, the research is of little value despite the statistics. In other words, statistical analysis can add to the precision and reliability of our research, but we need validity too. Both analytical rigour as well as technical strength are essential.

16.6 Annotated references

Cozby, P. (2009). Methods in Behavioral Research, 10th edition. Boston: McGraw Hill. A good clear psychology text with chapters on statistics.

Gaur, A. and S. Gaur. (2009). Statistical Methods for Practice and Research: A Guide to Data Analysis Using SPSS, 2nd edition. New Delhi: Response.Co

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This book takes you through using the SPSS package for the types of test outlined in this chapter and much more.

Israel, D. (2008). Data Analysis in Business Research: A Step-by-Step Nonparametric Approach. New Delhi: Response. Contains the major non-parametric statistical tests that can be used with small

samples in all social science subjects, assuming little knowledge of statistics.

Kanji, K. (2006). 100 Statistical Tests, 3rd edition. New Delhi: Vistaar. This book covers the most commonly used statistical tests, both parametric and

non-parametric. For each test, it describes the purpose, limitations and assumptions, a worked example and the calculation.

An internet search will help you obtain much guidance on using Excel for data analysis. There are also several books in the market showing how to use it.

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