Statistics Overview for Research Studies

Descriptive Statistics

• Relates to samples.
• Used to describe a sample.
• Purpose: external validity, generalizability of the study.
• Examples include age and sex distribution.

Inferential Statistics

• Relates to populations.
• Usually too expensive and time-consuming to measure the entire population.
• Use a sample to make inferences about the population.

Sampling Variation

• Sample mean is unlikely to be exactly equal to the population mean.
• Different samples would yield different estimates due to sampling variation.
• Sampling distribution of the mean – independent samples of the same size from the same population and their sample means.

Sampling Distribution and Standard Error

• Means from multiple samples of the same size from the same population form a normal distribution.
• The mean of this distribution is the population mean.
• The standard deviation of this distribution (standard error of the sample mean) equals the population standard deviation divided by the square root of the sample size.

Standard Error

• Measures the precision of estimating the population mean from a sample mean.
• Larger sample sizes result in smaller standard errors.
• Standard error depends on population variability.

Standard Deviation vs. Standard Error

• Standard deviation describes variation within a sample.
• Standard error describes variation between sample means.
• Standard deviation – how much individual observations vary from the sample mean.
• Standard error – how much mean scores from different samples vary from the true population mean.

Normal Distribution

• Variables approximate a normal (Gaussian) distribution.
• The normal curve is symmetrical and bell-shaped.
• The curve's width decreases as the standard deviation decreases.
• Examples of normally distributed variables include height, IQ, and blood pressure.
• Life expectancy is not normally distributed.

Central Limit Theorem

• The sampling distribution of the mean is normal, even if the original variable isn't.

Z-scores

• A way to standardize any normally distributed variable: subtract the mean and divide by the standard deviation..
• z-scores represent how far an observation is from the mean in standard deviation units.
• Allow the determination of the proportion of a population falling within a specified range.
• Any normally distributed variable can be converted to a standard normal distribution.
• Mean = 0 and standard deviation = 1.
• Subtract the mean from each observation and divide by the standard deviation.
• Any observation can be turned into a z-score.
• Measure of how far an individual score is from the mean.
• Measured in units of standard deviations (+1 means one standard deviation above the mean).
• Determines the proportion of a population with values in a specified range – used in confidence intervals and p-values.

Properties of the Normal Distribution

• Symmetrical; 50% of observations are above the mean, and 50% below.
• Mean = median = mode.
• Completely described by the mean and standard deviation.
• IQ: mean = 100; standard deviation = 15.
• Standard normal distribution: mean = 0; standard deviation = 1.
• Shape always the same.
• Specific percentages of data fall within specific intervals of standard deviations (e.g., 68.3% within ±1 SD, 95.5% within ±2 SD, 99.7% within ±3 SD).
• The tails of the distribution contain increasingly smaller proportions of data further from the mean.

Confidence Intervals

• 95% of sample means fall within ±2 standard errors of the population mean.

Reporting Results (Example)

• A significant difference was found between two treatments (p-value).
• We found a significant difference between the IPT and CBT treatments (p-value).