Research in Physical Education II: Sampling Techniques, Data and its Types, the Normal Probability Curve and Grading Scales, Statistical Processes and their Uses, Parametric and Non-Parametric Techniques, and Computer Applications including SPSS

Data are recorded observations. They are primary when collected by the researcher for the study (test scores, questionnaire responses) and secondary when taken from existing sources (records, published statistics); qualitative when descriptive (interview themes) and quantitative when numerical; discrete when counted in whole units (goals scored, number of push-ups) and continuous when measured on a scale that can take any value (time, distance, weight). S. S. Stevens (1946) defined four levels of measurement. Nominal data only classify (sex, position played, jersey numbers). Ordinal data rank without equal intervals (finishing places in a race, grades A to E). Interval data have equal intervals but no true zero (temperature in degrees Celsius, many psychological scale scores). Ratio data have equal intervals and a true zero, so ratios are meaningful (height, weight, time, distance, force): a jump of 6 m is twice one of 3 m. The level of measurement decides which statistics may be used, and data are collected by tests and measurements, questionnaires, interviews, observation and records.

The normal probability curve is a symmetrical, bell-shaped distribution in which the mean, median and mode coincide at the centre, the tails approach but never touch the baseline (it is asymptotic), and the total area under it represents 100 per cent of the cases. Many physical measurements, such as height and many fitness scores in large unselected groups, are approximately normal. Its key property is that fixed proportions of cases lie within fixed distances from the mean, measured in standard deviations (σ): about 68.26 per cent within ±1σ, about 95.44 per cent within ±2σ and about 99.73 per cent within ±3σ. So about 34.13 per cent lie between the mean and +1σ, about 13.59 per cent between +1σ and +2σ, and about 2.14 per cent between +2σ and +3σ. A distribution that departs from symmetry is skewed (positively, with a long right tail, or negatively), and one more peaked or flatter than normal is leptokurtic or platykurtic.

Standard scores place a raw score on the curve. The z-score is z = (X − M) ÷ σ, the number of standard deviations a score lies from the mean; its mean is 0 and its standard deviation 1. Because z-scores are negative below the mean and have decimals, they are often converted to T-scores, T = 50 + 10z, with mean 50 and standard deviation 10. For a shot-put mean of 10 m and standard deviation 1.5 m, a throw of 13 m gives z = (13 − 10) ÷ 1.5 = 2 and T = 50 + 10 × 2 = 70. Standard scores allow performances in different events, measured in different units (seconds, metres, repetitions), to be compared and added into a single battery score. Grading on the curve assigns grades by position: in a common five-grade scale based on the normal curve, the middle grade C covers the band from −0.5σ to +0.5σ (about 38.29 per cent of pupils), B and D the bands from ±0.5σ to ±1.5σ (about 24.17 per cent each) and A and E the extremes beyond ±1.5σ (about 6.68 per cent each). The six-sigma scale divides the range from −3σ to +3σ into six equal bands; and percentiles give the percentage of cases falling below a score, so a z of +1 is at about the 84th percentile.

Statistics organise, summarise, analyse and interpret data; they make large masses of scores intelligible, allow comparison, test hypotheses and support prediction and the construction of norms. Descriptive statistics summarise a set of data: measures of central tendency (the mean, the arithmetic average; the median, the middle score; the mode, the most frequent score), measures of variability (the range, the quartile deviation and the standard deviation, the square root of the mean of squared deviations from the mean) and measures of relationship (correlation). For the scores 2, 4, 4, 4, 5, 5, 7 and 9, the mean is 40 ÷ 8 = 5; the squared deviations 9, 1, 1, 1, 0, 0, 4 and 16 sum to 32; dividing by N gives a variance of 4 and a standard deviation of 2 (dividing by N − 1, as for a sample estimate, gives about 2.14). Pearson's product-moment correlation (r) measures linear relationship between interval or ratio variables and ranges from −1 to +1; Spearman's rank-order correlation, ρ = 1 − 6Σd² ÷ n(n² − 1), uses ranks. For six athletes ranked by two judges with Σd² = 14, ρ = 1 − 84 ÷ (6 × 35) = 1 − 84 ÷ 210 = 0.6. Correlation does not prove causation.

Inferential statistics draw conclusions about a population from a sample. Parametric techniques assume interval or ratio data, a roughly normal distribution in the population, homogeneity of variance and independent observations, and are more powerful when the assumptions hold. They include the t-test for comparing two means (independent samples, with df = n1 + n2 − 2, or paired samples from the same subjects tested twice, with df = n − 1: fifteen athletes tested before and after training give 14 degrees of freedom), analysis of variance (ANOVA) for three or more means (for four groups of ten, df between groups = k − 1 = 3 and df within groups = N − k = 36), analysis of covariance (ANCOVA), which adjusts post-test means for pretest differences, Pearson's r and regression. Non-parametric (distribution-free) techniques make fewer assumptions and suit nominal or ordinal data, small samples and skewed distributions: the chi-square test for frequencies (for a table of 3 rows and 4 columns, df = (3 − 1)(4 − 1) = 6), the Mann–Whitney U test (the non-parametric equivalent of the independent t-test), the Wilcoxon signed-rank test (for paired data), the Kruskal–Wallis test (for three or more independent groups), the Friedman test (for repeated measures) and Spearman's ρ.

Statistical packages carry out the calculations and reduce errors. SPSS (originally the Statistical Package for the Social Sciences, first released in 1968 and now owned by IBM) is the most widely used in physical education and the social sciences: data are entered in the Data View with each row a case and each column a variable, variables are defined in the Variable View (name, type, label, values, measurement level), and menus run descriptive statistics, t-tests, ANOVA, correlation, regression, non-parametric tests and charts, with output that must still be interpreted by the researcher. Other packages include R (free and open source), SAS, Stata and the analysis tools of Microsoft Excel. Microsoft Office supports the whole process: Word for the proposal and report with reference management, Excel for data entry, calculation and graphs, and PowerPoint for presentations. E-mail is used to correspond with supervisors, co-authors, journals and participants, with attention to data protection; search engines and scholarly databases support the literature search; and reference managers such as Zotero and Mendeley format citations. Online survey tools, wearable sensors and video-analysis software now collect data directly, which makes careful data management, backup and privacy protection part of research ethics.