Business Statistics and Research Methods

Central tendency, dispersion, skewness, correlation

Central tendency summarises a distribution with a single typical value. The mean is the sum of observations divided by their number, sensitive to every value and therefore pulled by extreme outliers. The median is the middle value when data are ordered, robust to outliers and appropriate for skewed distributions and ordinal data. The mode is the most frequent value, useful for categorical data and for identifying a common size or choice. Their relationship reveals shape: in a symmetric distribution they coincide, in a positively skewed distribution the mean exceeds the median which exceeds the mode, and in a negatively skewed distribution the order reverses. For grouped data, the mean uses class midpoints weighted by frequencies, the median uses cumulative frequencies to locate the median class, and the mode uses the class with the highest frequency, adjusted by neighbouring frequencies where required.

Dispersion measures spread around the centre, and without it a mean is incomplete. Range is the difference between maximum and minimum, simple but driven by extremes. Quartile deviation uses the interquartile range, the middle half of the data, and is robust to outliers. Mean deviation averages absolute deviations from mean or median. Standard deviation, the square root of average squared deviation from the mean, is the most widely used measure because it enters further statistical theory, and its square is variance. The coefficient of variation divides standard deviation by mean, allowing comparison of variability between series with different units or levels. A low coefficient means relative consistency, a high one means relative volatility, which matters when comparing, for example, the stability of sales across two very different product lines.

Skewness measures asymmetry. Positive skew has a long right tail, with most values low and a few high values pulling the mean up. Negative skew has a long left tail. Karl Pearson's coefficient uses the difference between mean and mode divided by standard deviation, while the quartile-based coefficient uses quartiles and is robust to extremes. Kurtosis measures peakedness relative to the normal distribution: leptokurtic distributions are peaked with heavy tails, platykurtic distributions are flat with light tails, and mesokurtic distributions match the normal shape. Correlation measures linear association between two variables. Karl Pearson's coefficient ranges from minus one to plus one, with sign showing direction and magnitude showing strength, while Spearman's rank correlation handles ordinal data and monotonic relationships by correlating ranks. Correlation does not prove causation: a strong correlation may reflect a common cause, reverse causation, or coincidence, and must be checked with a scatter diagram, which also reveals non-linear patterns and outliers that a single coefficient hides. Regression extends correlation to prediction, with the line of best fit estimated by least squares, slope showing the change in the dependent variable per unit change in the independent variable, and the coefficient of determination showing the proportion of variation explained.

Probability and probability distributions

Probability quantifies uncertainty on a scale from zero, impossible, to one, certain. Classical probability divides favourable outcomes by total equally likely outcomes, empirical probability uses observed relative frequency, and subjective probability expresses personal judgment where neither is available. The addition rule gives the probability of either of two events, subtracting their joint probability when they overlap, while mutually exclusive events cannot occur together so their joint probability is zero. The multiplication rule gives the probability of both events, using conditional probability when events are dependent: the probability of A and B equals the probability of A multiplied by the probability of B given A. Independent events satisfy the simpler product rule, and Bayes' theorem updates a prior probability with new evidence to give a posterior probability, a logic central to testing, screening and decision making under uncertainty.

A random variable assigns a number to each outcome. Discrete variables take countable values, continuous variables take values over an interval. Expected value is the probability-weighted average, and variance measures spread around it. The binomial distribution models a fixed number of independent trials with two outcomes and constant success probability, with mean equal to number of trials multiplied by probability and variance equal to trials multiplied by probability multiplied by one minus probability. It suits pass-fail, defective or not, and buy or not-buy situations. The Poisson distribution models rare events over time or space, such as arrivals or defects, with mean equal to variance, and approximates the binomial when trials are many and probability small. The normal distribution is the symmetric bell-shaped continuous distribution defined by its mean and standard deviation, with about two-thirds of values within one standard deviation, about nineteen-twentieths within two, and almost all within three. Standardisation converts any normal value to a standard normal score by subtracting the mean and dividing by standard deviation, allowing probabilities to be read from tables. The central limit theorem explains the normal distribution's importance: the mean of a large sample is approximately normal regardless of the population shape, which justifies much of inferential statistics.

Research, data and sampling

Research is systematic enquiry aimed at answering a question or solving a problem with evidence. Business research begins by defining the problem precisely: a vague aim such as improving sales must become a researchable question, such as which factors drive repeat purchase among a defined customer group. The literature review then shows what is already known and where the gap lies. Objectives and hypotheses follow, then research design. Exploratory design clarifies a poorly understood problem through literature, expert interviews and pilot work. Descriptive design measures characteristics, such as market size or customer profile, through surveys and observation. Causal design tests cause and effect through experiments, controlling other factors so that a change in the outcome can be attributed to the treatment. The design must match the question: a survey cannot prove causation as an experiment can, and an experiment on a tiny convenience sample cannot support broad generalisation.

Data are primary when collected first-hand for the study, through questionnaires, interviews, observation or experiments, and secondary when collected earlier for another purpose, from company records, government publications, industry reports or databases. Secondary data are cheaper and faster but may be outdated, defined differently, or biased for their original purpose, so they must be evaluated for relevance, accuracy and currency before use. Measurement requires validity, measuring what is claimed, and reliability, giving consistent results on repetition. Scales progress from nominal classification, through ordinal ranking, interval measurement with equal gaps but arbitrary zero, to ratio measurement with a true zero permitting all arithmetic. Questionnaires need clear wording, no leading or double-barrelled questions, sensible ordering, and pilot testing, while interviews trade depth against cost and interviewer bias, and observation avoids what respondents say in favour of what they do.

Sampling selects a subset to represent a population, because studying everyone is usually too costly or slow. Probability sampling gives every member a known, non-zero chance of selection. Simple random sampling gives equal chance, systematic sampling takes every nth member from a list, stratified sampling divides the population into homogeneous strata and samples within each, improving precision for subgroups, and cluster sampling selects whole groups, useful when the population is geographically spread. Non-probability sampling includes convenience, judgment, quota and snowball sampling, which are cheaper and faster but cannot support statistical generalisation, because selection bias is uncontrolled. Sampling error arises from studying a sample rather than the whole population and shrinks with sample size, while non-sampling error arises from faulty design, wording, non-response or recording, and does not shrink with size. A good sampling plan states the population, sampling frame, method, size and expected error before fieldwork begins.

Hypothesis testing and report writing

Hypothesis testing formalises a decision about a population claim using sample evidence. The null hypothesis states no effect, no difference or no association, and is the claim tested. The alternative hypothesis states the effect the researcher hopes to find, one-sided when direction is predicted, two-sided when any difference counts. The significance level, commonly set at five percent, is the probability of rejecting a true null hypothesis, a Type I error. A Type II error is failing to reject a false null hypothesis, and power is the probability of correctly rejecting a false null, which rises with sample size and effect size. The test statistic compares the observed sample result with the null value, scaled by standard error, and the p-value is the probability of a result at least as extreme as observed if the null were true. Reject the null when the p-value is below the significance level, or when the test statistic falls in the critical region, otherwise fail to reject it, a wording that deliberately avoids claiming the null is proved.

Choice of test depends on data and question. A one-sample test compares a mean or proportion with a known value. A two-sample test compares two groups, paired when the same subjects are measured twice, independent otherwise. The t-test is used for means with small samples or unknown population standard deviation, the z-test for large samples or known standard deviation, and the chi-square test for association between categorical variables and for goodness of fit, comparing observed frequencies with expected frequencies under the null. Analysis of variance compares means across more than two groups by splitting total variation into between-group and within-group components, with the F ratio testing whether group differences exceed random variation. Non-parametric tests, such as the Mann-Whitney test and Wilcoxon signed-rank test, are used when normality cannot be assumed or data are ordinal. Every test carries assumptions, normality, independence, equal variances or adequate expected frequencies, and violating them without a robust alternative weakens the conclusion. Statistical significance is not practical importance: a tiny effect can be significant with a huge sample, and a large effect can be non-significant with a tiny sample, so effect size and confidence intervals should always accompany a p-value.

A research report communicates the whole study so that a reader can judge and replicate it. It typically opens with a title page, executive summary and contents, then introduction and problem statement, literature review, objectives and hypotheses, research methodology including design, data, sampling and tools, data analysis with tables and interpretation, findings, conclusions and recommendations, limitations, and references with appendices for the questionnaire and detailed tables. Writing must be precise and honest: report negative results, state limitations such as small sample or non-response, distinguish findings from interpretation, and avoid overstating causation from correlational designs. Tables need titles and source notes, charts need labelled axes, and every claim in the conclusions should trace back to an analysed result. A report that hides its methods or inflates its findings fails the very purpose of research, however polished its presentation.

Measures of central tendency compared

Sampling methods compared