Summary of Research Methods and Statistical Concepts

## Introduction Descriptive and inferential statistics are two complementary branches of statistics used to summarize data and make conclusions beyond the observed sample. Descriptive statistics organize and describe data numerically and graphically. Inferential statistics use sample data to make decisions or predictions about a larger population. > **Definition:** Descriptive statistics summarize observed data using measures such as mean, median, mode, range, variance and standard deviation. Inferential statistics draw conclusions about populations using sample data and probability. ## Descriptive Statistics: Building Blocks ### 1. Measures of Central Tendency - **Mean (average):** Sum of all values divided by the number of values. Useful for interval and ratio data. - **Median:** Middle value when data are ordered. Resistant to extreme values. - **Mode:** Most frequent value(s). Applicable to all scales. > Example: For scores $10,12,15,18,20$, the mean is $\frac{10+12+15+18+20}{5}=15$. ### 2. Measures of Spread - **Range:** Difference between maximum and minimum. - **Variance:** Average squared deviation from the mean. For a sample: $s^2 = \frac{1}{n-1}\sum_{i=1}^n (x_i-\bar{x})^2$. - **Standard deviation:** Positive square root of variance: $s = \sqrt{s^2}$. It measures typical distance from the mean. > Example: If $\bar{x}=15$ and deviations squared sum to $40$ for $n=5$, then $s^2=\frac{40}{4}=10$, $s=\sqrt{10}$. ### 3. Shape of Distribution - **Symmetrical distribution:** Left and right mirror each other. Mean $=$ median $=$ mode. - **Positively skewed:** Long tail to the right. Mode < median < mean. - **Negatively skewed:** Long tail to the left. Mean < median < mode. Fun fact: The mean is pulled toward the tail in skewed distributions, so median often better represents the "middle" for skewed data. ## Inferential Statistics: Making Decisions - Uses probability models to generalize from a sample to a population. - Common procedures: confidence intervals, hypothesis tests, regression, and ANOVA. > **Definition:** Inferential statistics estimate population parameters and test hypotheses using sample data and probability theory. ### Key ideas 1. Sampling and randomness: Good samples should be randomly chosen (see Random assignment vs random sampling below). 2. Sampling distribution: Distribution of a statistic over many samples (central limit theorem makes sample means approximately normal for large n). 3. Confidence intervals: Range of plausible values for a population parameter. 4. Hypothesis testing: Procedure to assess evidence against a null hypothesis using a test statistic and p-value. Did you know that the central limit theorem implies sample means tend toward a normal distribution even when the underlying data are not normal, provided the sample size is sufficiently large? ## Important Concepts Explained (from the provided content) ### Random assignment > **Definition:** Random assignment is allocating participants to experimental groups by chance so that each participant has an equal probability of being in any group. - Purpose: Control for confounding variables and allow causal inference in experiments. - Example: Assigning students randomly to two teaching methods before measuring test performance. ### Independent variable > **Definition:** The independent variable is the factor manipulated or categorized to observe its effect on the dependent variable. - In experiments, the independent variable is controlled by the researcher (e.g., teaching method). - In observational studies, it is a predictor or grouping factor (e.g., age group). ### Inferential statistics (revisited) - Use sample data to make statements about population parameters with quantified uncertainty (e.g., "the average test score is between 70 and 75 with 95% confidence"). ### Negatively skewed distribution > **Definition:** A distribution with a long tail to the left; most values cluster on the right. - Order of central tendenc