Unit 5: Business Statistics and Research Methods mind map

Organising data, the three kinds of average, measures of spread, and the shape of a distribution.

Organising and presenting data

Processing and presenting data

Raw data is processed in a fixed order before it is analysed. Graphs and tables then present it.

• Order: editing, coding, classification, tabulation, using percentages.
• Data preparation includes coding, data entry and editing.
• An exclusive class has an upper limit that is the lower limit of the next class.
• An ogive is the graph of a cumulative frequency distribution.

Editing comes first because errors must be removed before anything else is done. A data array lists raw values in order, so it is not compressed data. A frequency distribution, histogram and ogive are compressed forms.

Data mining and retail analytics

Data mining uses search techniques to find patterns in large data sets.

• Uses: predicting trends, analysing customer demographics, credit risk analysis.
• It finds buying patterns in retail.
• Electronic data interchange is not a data mining use.

A shop may find that people who buy bread also buy butter. That is a pattern found by mining. Factor analysis and regression analysis are statistical methods, not search techniques.

Measures of central tendency

Arithmetic, geometric and harmonic mean

The three means suit different data. Arithmetic is the all-purpose mean. Geometric suits growth rates. Harmonic suits rates when the distance is fixed.

• Order: AM is at least GM, and GM is at least HM.
• GM squared = AM x HM for two numbers.
• Adding a constant to every value adds it to the mean.
• Harmonic mean fits speed when the distance is fixed and time varies.

A car goes 60 km at 30 km/h and returns at 60 km/h. The average speed is the harmonic mean, 40 km/h, not 45. Growth rates of 5, 12 and 7 per cent have a geometric mean just under 8 per cent.

Positional averages and the relation of mean, median and mode

The median and mode are positional averages. The mean, geometric and harmonic means are mathematical averages.

• Positional averages are not affected by extreme values.
• Median and mode are not capable of algebraic treatment.
• For a moderately skewed distribution: Mean - Mode = 3 (Mean - Median).
• Equivalently, Mode = 3 Median - 2 Mean.

The median is the middle value and does not care how big the largest value is. A trimmed mean and median are good when outliers are present. The intersection of the less-than and more-than ogives gives the median. If mean is 20 and median is 18, the mode is 3 x 18 - 2 x 20 = 14.

Measures of dispersion

Range, quartile deviation, mean deviation and standard deviation

Dispersion shows how spread out the data is. The standard deviation is the most used measure.

• Dispersion measures: range, mean deviation, quartile deviation, standard deviation, coefficient of variation.
• Quartile deviation = (Q3 - Q1) / 2.
• Normal data: MD is about 0.8 of SD. QD is about 0.6745 of SD.
• Coefficient of correlation is not a measure of dispersion.

In a symmetrical distribution, the median lies halfway between Q1 and Q3. If Q1 is 48 and QD is 6, then Q3 is 60 and the median is 54. The standard deviation of the first seven natural numbers is 2.

Properties of standard deviation and variance

The standard deviation does not change when a constant is added. It is multiplied when the data is multiplied.

• Independent of change of origin, not of change of scale.
• Multiplying by K multiplies SD by K and variance by K squared.
• The sum of absolute deviations is least from the median.
• Sample variance divides by n - 1 and uses the sample mean.

Adding 10 to every value shifts the set, but the spread is unchanged. Multiplying by 10 stretches the spread. Bessel's correction divides by n - 1 because deviations from the sample mean are slightly too small.

Coefficient of variation, z-score and the box plot

The coefficient of variation compares spread across data sets. The z-score tells how far a value is from the mean in SD units.

• CV = SD / mean x 100. It measures relative risk.
• Z = (x - mean) / SD.
• A box plot shows the minimum, Q1, median, Q3 and maximum.
• The five-number summary has the same five values.

CV has no units, so you can compare rupee incomes with marks. A lower CV means steadier data. A box plot does not show the mean or the mode. Z is not 'x divided by the mean'.

Skewness and kurtosis

Skewness: direction and measures

Skewness is lack of symmetry. The tail shows the direction.

• Positive skew: long right tail, mean above median above mode.
• Negative skew: long left tail, mean below median below mode.
• Karl Pearson's measure: (Mean - Mode) / SD, or 3 (Mean - Median) / SD.
• Bowley uses quartiles and Kelly uses percentiles.

Skewness and the measures are unchanged by adding a constant or dividing by a constant, because the numerator and SD scale together. In a positively skewed distribution, Q3 - Median is larger than Median - Q1.

Kurtosis and the beta coefficients

Kurtosis is peakedness. Beta 2 equals 3 for a normal curve.

• Beta 2 equals 3: mesokurtic (normal).
• Beta 2 above 3: leptokurtic (peaked).
• Beta 2 below 3: platykurtic (flat).
• Beta 1 equals zero for a symmetric distribution.

Do not mix skewness and kurtosis. Skewness is lopsidedness. Kurtosis is how tall and thin the curve is. In a moderately asymmetrical distribution, SD is about 1.25 times the mean deviation.

The rules of probability, Bayes' theorem, and the binomial, Poisson, normal and other distributions.

Probability rules

Addition and complement rules

The addition rule finds the chance of one event or another. Subtract the overlap so it is not counted twice.

• P(A or B) = P(A) + P(B) - P(A and B).
• For mutually exclusive events, P(A or B) = P(A) + P(B).
• Complementary events: probabilities add to 1. They are exclusive and exhaustive.

In a pack of cards there are 4 kings and 26 red cards. Two kings are red, so a king or a red card has 4 + 26 - 2 = 28 outcomes out of 52. That is 7/13.

Counting, dice and event types

Multiply the outcomes of separate experiments. Count favourable cases carefully.

• A coin and two dice: 2 x 36 = 72 outcomes.
• Two dice have 36 outcomes. A sum of 8 has 5 of them.
• Compound event: joint occurrence. Collectively exhaustive: all possible outcomes. Equally likely: none is preferred.

List the cases for a sum of 8: (2,6), (3,5), (4,4), (5,3), (6,2). That is 5/36. A leap year has 52 weeks and 2 extra days, among 7 possible pairs. Pairs with a Sunday or a Monday number 3, so the probability is 3/7.

Conditional probability, multiplication and Bayes' theorem

Conditional probability is the chance of one event given another. Bayes' theorem revises a prior probability when evidence arrives.

• P(E2 | E1) = P(E1 and E2) / P(E1).
• Multiplication: P(E1 and E2) = P(E1) x P(E2) for independent events.
• Bayes: P(Hi | E) = P(Hi and E) / P(E).
• Bayes steps: priors, conditional probabilities, joint probabilities, posterior.

The letter H stands for hypothesis. Bayes' theorem starts with prior probabilities of mutually exclusive and collectively exhaustive events. It then uses the conditional probabilities of the evidence to get the posterior.

Distributions

Random variables and the binomial distribution

A random variable takes values that depend on chance. A binomial distribution counts successes in n independent trials.

• A discrete random variable takes countable values.
• A continuous random variable takes any value in a range.
• Binomial: parameters n and p. Mean np, variance npq, SD root of npq.
• The binomial can be symmetric or skewed.

A discrete distribution assigns a probability to each possible value, and the probabilities add up to 1. When n is large and p is small, the binomial is approximated by the Poisson with mean np.

Poisson distribution

The Poisson distribution counts rare events in a fixed interval at a constant average rate. Its mean equals its variance.

• Formula: P(X = k) = e^(-lambda) x lambda^k / k!.
• Mean = variance = lambda.
• Examples: customers per hour, typing errors per page, accidents per day.
• Dice throwing is not a Poisson process.

If 2 customers arrive per minute on average, the chance of exactly 3 in a minute is 0.1353 x 8 / 6 = 0.1804. The chance of exactly 4 is 0.1353 x 16 / 24 = 0.0902. For lambda 5, SD is root 5 and CV is 44.7 per cent.

Normal distribution

The normal distribution is the bell-shaped curve. It has two parameters, the mean and the standard deviation.

• Symmetric, unimodal and continuous.
• Mean, median and mode are equal.
• It approaches the x-axis but never touches it.
• Total area is 1. Inflection at mean plus or minus one SD. Kurtosis is 3.

The standard normal has mean 0 and SD 1. A Z of 1.0 leaves 15.87 per cent in the upper tail. About 68.27 per cent lies within one SD of the mean, 95.45 within two and 99.73 within three.

Other distributions and their uses

Each distribution fits a situation. A match question pairs them.

• Exponential: time between arrivals.
• Pareto: ownership of income and property.
• Uniform: all values equally likely.
• Hypergeometric: sampling without replacement.

The Poisson counts how many events happen. The exponential measures the waiting time between them. The Pareto is a power law where a few hold a large share. The standard deviation of a uniform distribution is root of (b - a) squared / 12.

Expected value and the shape of sampling distributions

Expected value is the long-run average. The shapes of the t, z, chi-square, F and uniform distributions differ in symmetry.

• Expected value of a count = n x p.
• The standard normal and t are symmetric and bell-shaped.
• Chi-square and F are right-skewed.
• The uniform is flat.

In increasing order of symmetry the exam placed: uniform, chi-square, F, t and z. A point estimate of a mean is the sample mean. The mean of the sampling distribution of the mean equals the population mean.

How two variables move together: the correlation coefficient, the two regression lines, and goodness of fit.

Correlation

The coefficient of correlation

The correlation coefficient r measures the strength and direction of a linear link. It lies between -1 and +1.

• r = Cov(X, Y) / (SD of X x SD of Y).
• The sign gives direction. The size gives strength.
• r is independent of change of origin and scale.
• If X + Y is constant, r = -1.

A value of +0.87 means a strong positive relationship. A very high inverse relationship needs a value like -0.80. The number of pairs is not needed to find r. The denominator uses the product of SDs, not variances.

Probable error, rank correlation and measures of association

The probable error tests how reliable r is. The right measure of association depends on the level of measurement.

• PE = 0.6745 x (1 - r squared) / root n.
• Standard error of r = (1 - r squared) / root n.
• If |r| is more than 6 times PE, r is significant.
• Spearman's rho is for two ordinal variables.

Pearson's r and partial correlation suit interval and ratio data. Spearman's rho suits ranks. Phi and Cramer's V suit nominal data. The 0.6745 in PE is the value that cuts the middle half of a normal curve.

Regression

Regression coefficients and their properties

There are two regression lines, Y on X and X on Y. Their coefficients are linked to r.

• r = plus or minus the square root of (byx x bxy), the geometric mean.
• Both coefficients and r have the same sign.
• Both coefficients cannot be greater than 1 at the same time.
• The two lines meet at the point of the two means.

If byx is 0.8 and bxy is 1.2, r is the root of 0.96, about 0.97. If both coefficients are negative, r is negative. The product byx x bxy equals r squared, which cannot be negative. The coefficient of correlation is the geometric mean, not the harmonic mean.

Change of origin and scale

A change of origin does not alter regression coefficients. A change of scale does.

• Regression coefficient is independent of origin only.
• Multiply x by kx and y by ky: byx is multiplied by ky / kx.
• bxy is multiplied by kx / ky.
• r is unchanged. Covariance and SD change with scale.

Take byx = 2.4 and bxy = 0.4. Multiply x by 5 and divide y by 2. So kx = 5 and ky = 0.5. New byx = 2.4 x 0.5 / 5 = 0.24. New bxy = 0.4 x 5 / 0.5 = 4.

Goodness of fit and regression assumptions

R squared measures the share of variation explained. Adjusted R squared lets you compare models.

• R squared = 1 - SSE / SST = SSR / SST.
• SST = SSR + SSE.
• Adjusted R squared corrects for the number of variables and observations.
• Tolerance (1 - R squared) measures multicollinearity.

With SST of 15,730 and SSE of 1,530, R squared is 1 - 0.0972 = 0.9028. That is a good fit. Plain R squared never falls when a variable is added. Errors in simple regression are assumed independent, not identical. R squared is usually smaller for cross-section data than for time series.

How a sample is chosen, the errors that can creep in, and how a sample statistic behaves.

Sampling techniques

Probability and non-probability sampling

In probability sampling every unit has a known, non-zero chance. In non-probability sampling, the researcher chooses.

• Probability: simple random, systematic, stratified, cluster, multi-stage.
• Non-probability: quota, convenience, purposive (judgement), snowball.
• Snowball suits hard-to-reach groups.
• Convenience is picking friends and neighbours.

Quota sampling fixes how many respondents are needed from each group, then fills them with whoever is convenient. There is no randomisation at selection, so it is non-random. Stratified sampling suits a population that is not homogeneous.

The sampling design process

Sampling design moves from who you study to how many you need.

• Define the target population.
• Decide the parameters of interest.
• Select the sampling frame, then the technique.
• Decide the sample size, then carry it out.

The sampling frame is the actual list from which units are drawn, such as an electoral roll. The size comes after the method. Different papers word the steps differently, but the order stays the same.

Sampling and non-sampling errors

Sampling error arises because only a part of the population is observed. Non-sampling errors come from mistakes in collecting and handling data.

• Non-sampling errors occur even in a census.
• Causes: vague definitions, defective method, incomplete coverage, wrong entries.
• A value from a sample is a statistic. A value from a population is a parameter.

Observing only a part and avoiding a census is the reason for sampling error, not non-sampling error. Statistical regularity and the law of large numbers support sampling.

Choosing a sampling method

Each method suits a situation. A match question gives the situation.

• Multi-stage: a widely spread population.
• Systematic: elements arranged in order, pick every kth item.
• Judgement: a few respondents best placed to give the information.
• Stratified: heterogeneous sub-populations.

A small sample from an unknown population standard deviation needs the t-test, not the z-test. Mixing these up is a common trap.

Sampling distributions

Standard error of the mean

The standard error is the standard deviation of the sampling distribution. It falls as the sample grows.

• SE = sigma / root n.
• Without replacement: multiply by the root of (N - n) / (N - 1).
• SE to sigma ratio = 1 / root n.
• About 68.26 per cent of sample means fall within one SE.

If the population SD is 50 and n is 100, SE is 5. The chance that the sample mean is within 5 of the population mean is the chance of being within one SE, 0.6826. If SE : sigma is 8 : 40, that is 1/5, so root n is 5 and n is 25.

Properties of a good estimator

A good point estimator is unbiased, consistent and efficient.

• Unbiased: its expected value equals the parameter.
• Consistent: it gets closer to the parameter as the sample grows.
• Efficient: the smallest variance among unbiased estimators.
• Sufficient: it uses all the information in the sample.

Consistency is a large-sample idea. Unbiasedness is about the average. Stationarity and neutrality are not properties of estimators. Chi-square, F and uniform distributions are less symmetric than the t and z distributions.

The steps of a test, the two errors, which test fits which situation, and the critical values.

Steps, errors and critical values

Steps in hypothesis testing

A test follows a fixed order. Hypotheses come first and the conclusion comes last.

• Set the null and alternative hypotheses.
• Choose the level of significance.
• Select the test statistic.
• Set the decision rule, compute, then draw a conclusion.

The null hypothesis is the no-effect statement. The level of significance is chosen before seeing the data. The decision rule uses the critical value or the rejection region.

Type I and Type II errors, power and p-value

A Type I error rejects a true null hypothesis. A Type II error fails to reject a false one.

• Type I: false alarm, probability alpha.
• Type II: missed detection, probability beta.
• Power = 1 - beta.
• Effect size measures the magnitude of an effect, regardless of sample size.

In the story of Dushyant and Shakuntala, the true statement 'she is my wife' was rejected because the ring was missing. That is a Type I error. A p-value is the smallest level at which the null can be rejected. A large sample can make a trivial effect significant, so effect size matters.

Critical values of Z

Critical values of Z depend on the level of significance and on the tails of the test.

• Two-tailed 5 per cent: plus or minus 1.96.
• Two-tailed 1 per cent: plus or minus 2.575.
• One-tailed 5 per cent: 1.645.
• One-tailed 1 per cent: 2.33.

A one-tailed test puts the whole alpha in one tail. A test with H1: mu1 greater than mu2 is right-tailed. At 10 per cent in one tail, Z is about 1.28. Two-tailed tests split alpha between the tails, so each tail holds half.

Choosing the test

Z, t, F and chi-square tests

Each test answers a particular question. Choose by data type and sample size.

• Z-test: means of large samples.
• t-test: means of small samples, or paired samples.
• ANOVA (F-test): means of more than two groups.
• Chi-square: association between attributes and goodness of fit.

The paired t-test compares the same subjects twice. The F-test compares more than two groups, with the same assumptions as t: normality, independence and equal variances. The F value is never negative, because it is a ratio of variances. The critical value of t is always larger than that of z for the same n and alpha, and it falls as n rises.

Chi-square test

The chi-square test works on counts. It is non-parametric.

• Test statistic = sum of (O - E) squared / E.
• It can never be negative. It is continuous.
• Its only parameter is the degrees of freedom.
• The data must be counts, not percentages.

Chi-square was developed by Karl Pearson, not Spearman. If expected frequencies are too small, the value is overestimated and the test rejects too often. A large value means a poor fit.

Parametric and non-parametric tests

Non-parametric tests do not assume a normal population. Each is the rank-based twin of a parametric test.

• Mann-Whitney U: independent t-test.
• Wilcoxon signed rank: paired t-test.
• Kruskal-Wallis: one-way ANOVA.
• Friedman: two-way ANOVA.

The Wald-Wolfowitz runs test has no parametric counterpart. Mann-Whitney needs independent samples and is used when normality is not assumed. Chi-square, Mann-Whitney and Kruskal-Wallis are non-parametric. The F-test and t-test are parametric.

Test statistics and their formulas

Each test has its own statistic. A match question gives the formula and asks for the test.

• Chi-square: sum of (O - E) squared / E.
• Z-test: (sample mean - population mean) / standard error.
• F-test: mean square between / mean square within.
• H-test (Kruskal-Wallis) uses rank sums R.

The giveaway for Kruskal-Wallis is the letter R, which stands for ranks. Only non-parametric tests use ranks. The standard error for Z is sigma over root n.

One-factor ANOVA

One-factor ANOVA compares the means of more than two groups. The F value is a ratio of variances.

• F = mean square between / mean square within.
• F can never be negative.
• It does not need equal group sizes, but unequal sizes reduce power.
• t and F tests rest on the same assumptions.

Both numerator and denominator are variances built from squares, so F cannot be negative. The t-test compares two groups. The F-test compares more than two. The assumptions are normality, independence and equal variances.

How a study is planned and run: the research process, design types, experiments, data collection, measurement and report writing.

Research process and design

The research process and problem definition

Research follows a logical order, from a clear problem to a written report. Each step prepares the next.

• Define the problem, formulate hypotheses, prepare the research design.
• Collect and analyse data, interpret and write the report.
• A research proposal adds a literature review after the problem is defined.
• Problem definition goes from key problems to a research question and hypothesis.

The order differs slightly by source. Marketing research puts problem definition, then approach, then design, then data and then the report. For a proposal, the order is problem, literature review, hypothesis, design, data collection. In the scientific method, observation leads to induction, explanation, prediction and testing.

Research design and its dimensions

A research design is the framework for collecting and analysing data. It can be classified in several ways.

• Time dimension: cross-sectional (one point in time) or longitudinal (repeated).
• Purpose: exploratory, descriptive or experimental.
• Experimental is the most precise design.
• A longitudinal study measures changing opinions again and again.

An exploratory study is finished when the researcher knows the major dimensions of the task, has subsidiary investigative questions and a set of hypotheses. A research design is not chosen by Cronbach's alpha. That measures reliability.

Experimental designs

In an experiment the researcher manipulates an independent variable and watches the dependent variable. Designs differ in control.

• Experimental treatment: manipulating the independent variable.
• Pre-experimental designs: one-group pre-test post-test, static group comparison, after-only study.
• Complete designs: factorial, Latin square, before-after with control groups.
• One-group before-after is incomplete, because it has no control group.

Without a control group, any change might be caused by something else. A factorial design tests several factors and their interactions. A well-planned experiment: choose variables, set treatment levels, choose a design, assign subjects and pilot test, and control extraneous factors.

Research questions, spurious relationships and variables

Management problems are turned into research questions in steps. A spurious relationship is one that only appears to be real.

• Hierarchy: management dilemma, management question, research question, investigative question, measurement question.
• Measurement questions are the ones actually asked of respondents.
• Spurious: each variable is related to a third variable.

Ice-cream sales and drowning rise together, but hot weather causes both. The independent variable is the cause. The dependent variable is the effect.

Stages of a research project and qualitative research

Stages of a project run from planning and budgeting to fieldwork and the report. Qualitative research has its own order.

• Project stages: budgeting, field work, data collection, outcomes, report writing.
• Qualitative research: topic, literature review, purpose and participants, data collection and analysis, report.
• An investigation begins with instructions from the client and terms of reference.

Budgeting comes first because nothing can be commissioned without money. In qualitative research, a clear topic sets the direction, and the literature review sharpens the question. An investigation then plans the work, collects documents and removes inconsistencies by calculation.

Data collection and measurement

Primary and secondary data

Primary data is collected fresh for the study. Secondary data already exists.

• Primary: interview, questionnaire, observation.
• Secondary: published reports, annual reports, unpublished theses.
• Consumer interviews have problems: non-random samples and response bias.

Interviews, questionnaires and observation are first-hand methods, so they give primary data. An unpublished thesis and an annual report already exist, so they are secondary. The identification problem is not a main weakness of consumer interviews.

Scales of measurement

Data are measured on four scales, from weakest to strongest. Each adds properties.

• Nominal: numerals only as labels.
• Ordinal: ranks.
• Interval: equal gaps, no true zero.
• Ratio: equal gaps and a true zero.

Assigning numerals to objects to represent their attributes is nominal data. Interval and ratio data suit parametric tests and Pearson's r. Ordinal data suit rank methods. Factors that decide the choice of a scale: research objectives, data properties and the number of dimensions.

Questionnaire design and reliability

A good questionnaire controls bias. Reliability and validity check the measure.

• Order bias is reduced by funnel technique, filter questions and pivot questions.
• A filter question checks whether the respondent knows the subject.
• Leading questions push the respondent to a particular answer.
• Cronbach's alpha measures internal consistency, which is reliability.

The funnel technique starts broad and narrows down. Filter questions skip those who do not qualify. A low alpha means poor reliability, not good. A high Cronbach's alpha shows that items measure the same thing.

Report writing

Components and order of a research report

A report has prefatory, main and supplementary parts. The executive summary sits in the prefatory part.

• Prefatory: letter of transmittal, title page, authorization statement, executive summary, contents.
• Introductory: problem definition, background, scope.
• Main body: research design, findings.
• Supplementary: glossary, appendices, bibliography.

Before writing, a researcher considers the purpose of the study, the readers and the uses of the report. The letter of transmittal comes first. The executive summary comes before the table of contents.