Statistics in Psychology: Descriptive, Inferential, and Distribution Analysis

Descriptive statistics

Statistics used to organize, summarize, and describe a set of data.

Inferential statistics

Statistics used to make guesses or inferences about a larger population based on a sample.

Population

The entire group of individuals or scores you are interested in.

Sample

A smaller subset of the population that is actually measured.

Parameter

A value that describes a population, usually symbolized with Greek letters like μ or σ.

Statistic

A value that describes a sample, usually symbolized with letters like M or s.

Independent variable

The variable that is manipulated or used to define groups/conditions.

Dependent variable

The variable that is measured as the outcome.

Random sampling

A method where every member of the population has an equal chance of being selected.

Representative sample

A sample that accurately reflects the population it came from.

Sampling error

The natural difference between a sample statistic and the true population parameter.

Nominal level of measurement

Categories with no meaningful order, such as major, favorite color, or condition.

Ordinal level of measurement

Categories with a meaningful order, but unequal or unknown distances between ranks.

Interval level of measurement

Numbers with equal intervals but no true zero, such as temperature in Fahrenheit.

Ratio level of measurement

Numbers with equal intervals and a true zero, such as height, weight, or reaction time.

Frequency distribution

A table or graph showing how often each score or category occurs.

Simple frequency

The number of times a score or category appears.

Relative frequency

The proportion or percentage of times a score or category appears.

Histogram

A graph used for quantitative data where bars touch because the values are continuous.

Bar graph

A graph used for categorical data where bars do not touch.

Normal distribution

A symmetrical, bell-shaped distribution where most scores are near the center.

Skewed distribution

A distribution where scores pile up on one side and the tail stretches toward the other side.

Positive skew

A distribution with a tail extending to the right; mean is usually greater than median.

Negative skew

A distribution with a tail extending to the left; mean is usually less than median.

Bimodal distribution

A distribution with two clear peaks or modes.

Mean

The arithmetic average; add all scores and divide by the number of scores.

Median

The middle score when scores are ordered from smallest to largest.

Mode

The most frequently occurring score or category.

When is the mean best?

Use the mean for roughly normal quantitative data without extreme outliers.

When is the median best?

Use the median when a distribution is skewed or has extreme outliers.

When is the mode best?

Use the mode for nominal/categorical data or when you want the most common value.

Main difference between mean and median

The mean uses every score and is affected by outliers; the median only depends on the middle position.

Range

The highest score minus the lowest score.

Interquartile range / IQR

The range of the middle 50% of scores.

Variability

How spread out or different the scores are from each other.

Sum of Squares / SS

The sum of squared deviations from the mean.

Variance

The average squared distance from the mean.

Standard deviation

The typical distance scores fall from the mean.

Relationship between variance and standard deviation

Standard deviation is the square root of variance.

Population variance formula

σ² = SS / N.

Sample variance formula

s² = SS / (n - 1).

Why does sample variance use n - 1?

Because using n - 1 corrects bias and gives a better estimate of population variance.

Degrees of freedom / df

The number of values free to vary after restrictions are placed on the data.

Raw score / X

The original score before transformation.

z-score

A standardized score showing how many standard deviations a raw score is from the mean.

z-score formula

z = (X - μ) / σ.

Positive z-score

The raw score is above the mean.

Negative z-score

The raw score is below the mean.

z-score of 0

The raw score is exactly equal to the mean.

Standardized distribution

A distribution transformed into z-scores, usually with mean 0 and standard deviation 1.

Why are z-scores useful?

They allow scores from different distributions or scales to be compared.

Standard normal distribution

A normal distribution of z-scores with mean 0 and standard deviation 1.

68-95-99.7 rule

In a normal distribution, about 68% of scores fall within 1 SD, 95% within 2 SD, and 99.7% within 3 SDs.

Probability in statistics

The likelihood that a particular outcome or more extreme outcome will occur.

Distribution of sample means / DSM

A distribution made of sample means from all possible samples of the same size.

Sampling distribution

A distribution of statistics, such as means, from many samples.

Expected value of M

The mean of the distribution of sample means; usually equal to μ.

Standard error of the mean / σM

The standard deviation of the distribution of sample means.

Standard error formula when σ is known

σM = σ / √n.

What happens to standard error as n increases?

Standard error gets smaller because larger samples are usually more accurate.

Central Limit Theorem / CLT

As sample size increases, the distribution of sample means becomes normal, even if the population is not normal.

Law of large numbers

As sample size increases, sample statistics tend to get closer to the population parameter.

Hypothesis testing

A statistical process for deciding whether a sample result is extreme enough to reject the null hypothesis.

Null hypothesis / H0

The hypothesis that there is no effect, no difference, or no relationship.

Alternative hypothesis / H1

The hypothesis that there is an effect, difference, or relationship.

Alpha level / α

The cutoff probability for deciding a result is statistically significant, often .05.

p-value

The probability of getting the observed result, or something more extreme, if the null hypothesis is true.

Statistical significance

A result is statistically significant when p ≤ α, meaning it is unlikely under the null hypothesis.

Reject H0

Decision made when p ≤ α; the result is statistically significant.

Fail to reject H0

Decision made when p > α; there is not enough evidence for statistical significance.

Region of rejection

The area of the sampling distribution where results are extreme enough to reject H0.

Region of acceptance

The area of the sampling distribution where results are not extreme enough to reject H0.

One-tailed test

A hypothesis test where the alternative predicts a specific direction.

Two-tailed test

A hypothesis test where the alternative predicts a difference but not a specific direction.

Type I error

Rejecting a true null hypothesis; false positive.

How do you reduce Type I error?

Use a smaller alpha level, such as .01 instead of .05.

Type II error

Failing to reject a false null hypothesis; false negative.

Power

The probability of correctly rejecting a false null hypothesis; power = 1 - β.

How do you increase power?

Increase sample size, reduce variability, use a stronger design, or use a larger alpha level.

Effect size

A measure of how large or meaningful an effect is.

Cohen's d

An effect size that describes how many standard deviations apart a mean is from a comparison value or another mean.

Cohen's d rule of thumb: small effect

d ≈ 0.20.

Cohen's d rule of thumb: medium effect

d ≈ 0.50.

Cohen's d rule of thumb: large effect

d ≈ 0.80.

Confidence interval

A range of plausible population parameter values consistent with the sample statistic.

What does a 95% confidence interval mean?

The interval gives a range of plausible population values based on the sample and method used.

Confidence interval general formula

Sample statistic ± margin of error.

Margin of error

The critical value multiplied by the standard error.

What does a confidence interval tell you that a hypothesis test does not?

It estimates the plausible size/location of the population parameter, not just whether there is significance.

One-sample z-test

A hypothesis test used to compare one sample mean to a known population mean when σ is known.

When do you use a one-sample z-test?

Use it when comparing one sample mean to a population mean and the population standard deviation is known.

One-sample z-test formula

z = (M - μ) / σM.

H0 for one-sample z-test

μ = a specific population mean.

H1 for two-tailed one-sample z-test

μ ≠ a specific population mean.

One-sample t-test

A hypothesis test used to compare one sample mean to a known or hypothesized population mean when σ is unknown.

When do you use a one-sample t-test instead of z-test?

Use t when the population standard deviation is unknown and must be estimated with sample standard deviation.

Estimated standard error of the mean / sM

The estimated standard deviation of the sampling distribution when σ is unknown.

Estimated standard error formula

sM = s / √n.

One-sample t-test formula

t = (M - μ) / sM.

df for one-sample t-test

df = n - 1.