Descriptive statistics turn raw business data into a clear picture. This chapter, from a business statistics course, covers displaying data with graphs, locating data with quartiles and percentiles, and measuring the centre with the arithmetic mean, median, mode, and the geometric mean used for average growth rates, along with skewness and measures of spread.
Displaying and Locating Data
A statistical graph helps reveal the shape of a data set and is often clearer than a mass of numbers. After displaying data, measures of location describe where particular values sit within it. Quartiles divide ordered data into four equal parts, and percentiles mark the value below which a given percentage of the data falls. In business these locate a figure against its peers, such as where a store’s sales rank among all branches.
Measures of the Centre
The centre of a data set can be summarised in several ways. The arithmetic mean, written with sigma notation, adds all values and divides by their count. The median is the middle value once data is ordered, and the mode is the most frequent value. This business-focused treatment also introduces the geometric mean, which multiplies the values and takes a root; it is the correct average for rates that compound, such as annual growth rates or investment returns, where the arithmetic mean would overstate the result.
Skewness and Spread
A distribution is symmetric when its values are balanced around the centre, and skewed when one tail is longer, which pulls the mean away from the median. Comparing the mean, median, and mode indicates the direction of skew. Spread is measured by the range, the difference between the largest and smallest values, and by the variance and its square root, the standard deviation, which describe how far values typically lie from the mean. Together these show both the shape and the variability of business data.
The chi-square distribution is used to test claims about categorical data and variances. This chapter describes the properties of the chi-square distribution and applies it to goodness-of-fit tests, tests of independence between two variables, tests of homogeneity across populations, and tests about a single population variance.
Facts About the Chi-Square Distribution
The chi-square distribution is a family of right-skewed curves that take only non-negative values, because the statistic is built from squared quantities. Its shape depends on a single degrees-of-freedom value, and as that value grows the curve becomes more symmetric and bell-like. The distribution is used to compare observed counts with the counts expected under a hypothesis, and to draw conclusions about categorical data and about the variance of a population.
Goodness-of-Fit and Independence Tests
A goodness-of-fit test checks whether observed frequencies across categories match a claimed distribution, such as whether lottery numbers occur equally often. A test of independence uses a contingency table to decide whether two categorical variables are related or independent. In each case the test compares observed counts with expected counts, sums the standardised squared differences into a chi-square statistic, and compares it with the distribution to obtain a p-value and a decision.
Homogeneity and Single-Variance Tests
A test of homogeneity checks whether two or more populations share the same distribution across categories, extending the same counting logic to several groups at once. The chi-square distribution is also used to test a claim about a single population variance or standard deviation. Across all these uses the pattern is the same: measure how far the data depart from what a hypothesis predicts, express that gap as a chi-square value, and judge whether it is larger than chance would explain.
Linear regression and correlation describe how two numeric variables are related. This chapter reviews linear equations and scatter plots, shows how to fit a line of best fit through data using the regression equation, and explains how the correlation coefficient measures the strength and direction of a linear relationship.
Scatter Plots and Linear Equations
A relationship between two numeric variables is first explored with a scatter plot, which plots paired observations as points. If the points cluster around a straight line, the variables have an approximately linear relationship. A linear equation describes such a line through its slope and intercept: the slope gives the change in one variable for each unit change in the other, and the intercept gives the value where the line meets the vertical axis. The scatter plot shows whether a line is a reasonable summary.
The Regression Equation
The regression equation gives the line of best fit, the straight line that comes closest to all the points by making the total squared vertical distance from the points to the line as small as possible. This least-squares line can be used to predict the value of the dependent variable from a value of the independent variable. Predictions are most reliable within the range of the observed data, and extending the line far beyond that range can give misleading results.
The Correlation Coefficient
The correlation coefficient measures the strength and direction of the linear relationship between two variables. It ranges from negative one to positive one: values near positive one indicate a strong upward relationship, values near negative one a strong downward one, and values near zero little or no linear relationship. A value close to the extremes means the points lie near the line, while outliers, points far from the pattern, can distort both the line and the coefficient.
Many questions compare two groups rather than one. This chapter extends hypothesis testing to two samples, showing how to test the difference between two population means when standard deviations are known or unknown, how to compare two population proportions, and how to handle matched or paired samples where observations come in pairs.
Comparing Two Population Means
When two groups are compared, the hypotheses concern the difference between their population means. If the population standard deviations are known, the normal distribution is used; if they are unknown and estimated from the samples, the Student t distribution is used instead. The test statistic measures how far the observed difference between the two sample means lies from the value stated in the null hypothesis, usually a difference of zero, relative to the variability of the data.
Comparing Two Proportions
To compare two population proportions, such as the success rates of two treatments, the difference between the two sample proportions is examined. Under the assumption that the proportions are equal, the sampling distribution of their difference is approximately normal for large samples, so a z-based test applies. As with means, the observed difference is standardised and turned into a p-value that measures how surprising the result would be if the two proportions were truly equal.
Matched or Paired Samples
Sometimes the two samples are not independent but naturally paired, such as the same subjects measured before and after a treatment. In this case the analysis works with the differences within each pair rather than the two groups separately. The set of paired differences is then treated as a single sample, and a one-sample test on those differences checks whether the average change differs from zero, controlling for variation between individuals.
Hypothesis testing uses sample data to decide between two competing claims about a population. This chapter explains how to state null and alternative hypotheses, the difference between Type I and Type II errors, and how to carry out and interpret a test for a single population mean or proportion using a p-value.
Null and Alternative Hypotheses
A hypothesis test begins with two opposing statements about a population. The null hypothesis is the claim of no effect or no difference, taken as true unless the evidence contradicts it. The alternative hypothesis is what the researcher suspects may be true instead, and it may be one-sided or two-sided depending on the question. The test uses sample data to decide whether there is enough evidence to reject the null hypothesis in favour of the alternative.
Type I and Type II Errors
Because a decision is made from a sample, it can be wrong in two ways. A Type I error occurs when a true null hypothesis is rejected, and its probability is the significance level chosen for the test. A Type II error occurs when a false null hypothesis is not rejected. Lowering the chance of one type of error tends to raise the other, so the significance level is set in advance to balance the risks according to the consequences of each mistake.
Carrying Out the Test
To test a claim about a single population mean or proportion, the appropriate distribution is chosen: the normal distribution when the population standard deviation is known, and the Student t distribution when it is estimated from the sample. A test statistic is calculated and converted into a p-value, the probability of results at least as extreme as those observed if the null hypothesis were true. If the p-value is smaller than the significance level, the null hypothesis is rejected.
