10.2 Basic Statistical Concepts

Statistics are the backbone of marketing research. They help us make sense of data and draw meaningful conclusions. This section covers key concepts like descriptive and inferential statistics, measures of central tendency, and dispersion.

We'll also dive into probability and hypothesis testing. These tools allow researchers to make predictions, test assumptions, and determine the significance of their findings. Understanding different types of variables and statistical tests is crucial for choosing the right analysis method.

Fundamental Statistical Concepts

Descriptive vs inferential statistics

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• Descriptive statistics summarize and describe the basic features of a dataset without drawing conclusions beyond the data itself (mean, median, mode, standard deviation)
• Inferential statistics use sample data to make inferences or predictions about a larger population through hypothesis testing and estimation of population parameters, allowing for generalizing findings from a sample to a population

Measures of central tendency

• Mean represents the arithmetic average of a set of values, calculated as $\frac{\sum_{i=1}^{n} x_i}{n}$, sensitive to extreme values (outliers)
• Median is the middle value when a dataset is ordered from lowest to highest, robust to outliers and preferred for skewed distributions
• Mode is the most frequently occurring value in a dataset, can have no mode or multiple modes
• Measures of dispersion quantify the spread or variability of a dataset
  • Range is the difference between the maximum and minimum values, calculated as $\text{max}(x) - \text{min}(x)$, sensitive to outliers
  • Variance measures how far values are from the mean, calculated as $\frac{\sum_{i=1}^{n} (x_i - \bar{x})^2}{n-1}$, average squared deviation from the mean
  • Standard deviation is the square root of the variance, $\sqrt{\frac{\sum_{i=1}^{n} (x_i - \bar{x})^2}{n-1}}$, measures the spread of data in the same units as the original data

Probability and Hypothesis Testing

Probability and hypothesis testing

• Probability expresses the likelihood of an event occurring as a value between 0 and 1, following basic rules such as the addition rule for mutually exclusive events and multiplication rule for independent events
• Sampling distributions describe the distribution of a sample statistic (sample mean) over repeated samples, with the Central Limit Theorem stating that as sample size increases, the sampling distribution of the mean approximates a normal distribution
• Hypothesis testing involves the null hypothesis ($H_0$) of no effect or difference, alternative hypothesis ($H_a$ or $H_1$) of an effect or difference, p-value as the probability of observing a test statistic as extreme as the one calculated assuming $H_0$ is true, and significance level ($\alpha$) as the predetermined probability threshold for rejecting $H_0$ (0.01, 0.05, 0.10), leading to rejecting or failing to reject $H_0$ based on the p-value and $\alpha$

Variables and statistical tests

• Types of variables include nominal (categorical without inherent order, gender), ordinal (categorical with inherent order, survey responses), interval (numeric with equal intervals but no true zero, temperature in ℃), and ratio (numeric with equal intervals and a true zero, height)
• Appropriate statistical tests depend on the variable types and research question
  • Chi-square test assesses the association between two categorical variables
  • t-test compares means between two groups
    1. Independent samples t-test for comparing means from two independent groups
    2. Paired samples t-test for comparing means from two related groups or repeated measures
  • ANOVA (Analysis of Variance) compares means across three or more groups
  • Correlation measures the strength and direction of a linear relationship between two continuous variables
  • Regression models the relationship between a dependent variable and one or more independent variables