Interpreting Statistics

Core Principles in Mental Health Research: Interpreting Statistics

• Learning Outcomes: Understand and explain sampling variety; list key properties of sampling distributions; understand how these properties allow conclusions about populations; explain relationships between standard deviation (SD) and standard error (SE); calculate and interpret a 95% confidence interval (CI); calculate and interpret a p-value.

Sampling Variability & Standard Error

• Populations & Samples: Statistics describe sample characteristics (e.g., mean). Statistics estimate parameters. Parameters describe population characteristics. Statistics are needed to describe a population.

• Descriptive Statistics & Inferential Statistics:

  • Descriptive Statistics: Relates to samples; used to describe a sample, and its purpose is external validity; generalizability, examples like age & sex distribution of participants. It relates to populations; estimates are usually too expensive and time-consuming for entire populations.

  • Inferential Statistics: Uses samples to make inferences about populations.

• Sampling Variation: Sample mean is unlikely to be exactly equal to the population mean. Different samples lead to different estimates, this is due to sampling variation. Calculating a frequency distribution of sample means, is called the sampling distribution of the mean.

• Sampling Distribution & Standard Error: The means would form a normal distribution. The standard deviation of this distribution equals the population standard deviation divided by the square root of sample size. This is called the standard error (SE) of the sample mean.

  • SE = σ / √n (σ = population standard deviation, n = sample size)

Standard Deviation (SD) and Standard Error (SE)

• Standard Deviation (SD): Average difference between individual observations and the sample mean.

• Standard Error (SE): Average difference between sample means and the true population mean

• The Standard Error: Measures how precisely the population mean is estimated from sample means. The size of the SE depends on population variation and sample size. Larger sample sizes result in smaller SEs.

Standard Normal Distribution

• A change of units transforms any normally distributed variable into a standard normal distribution. The mean is 0, and the standard deviation is 1. This is done by subtracting the mean from each observation and dividing by the standard deviation.

Z-Scores

• Z-score is a measure of how many standard deviations a given observation is from the mean. Z = (x - μ) / σ (x = observation, μ = population mean, σ = population standard deviation)

Areas Under the Normal Distribution Curve

• The total area is 1 or 100%. Probability of an observation falling within a range can be determined using the Normal distribution curve. Calculations are aided by tables or computers for ranges based on z-scores.

Properties of the Normal Distribution

• Symmetrical, completely described by Mean & SD, shape remains constant. 68.3% of observations fall within ±1 SD; 95.5% within ±2 SD; and 99.7% within ±3 SD.

Confidence Intervals & P-Values

• Confidence Intervals: In sampling distributions of means, 95% of sample means lie within +/- 1.96 SE of the population mean. This is written as: μ is in the range x - (1.96 x SE) to x + (1.96 x SE). A 95% Confidence Interval (CI): Estimated mean +/- 1.96 SE

• P-Values: The probability of observing a difference at least as large as that in a sample, given there's no true effect in the population.

Hypothesis Testing

• Test the hypothesis, calculate the difference between groups, specify the null value, find the standard error and p-value