Statistics

Define statistics

The science dealing with the collection; analysis; interpretation; and presentation of numerical data.

What are Independent Variables (IVs)?

Factors manipulated by the researcher; assumed to cause changes in the dependent variable.　x

What are Dependent Variables (DVs)?

Factors that are measured; they change in response to the independent variable. y

What are Extraneous Variables?

Unwanted factors that can influence the relationship between IV and DV. Must be controlled or randomized.

Name the four components of scientific research

Objectivity; Confirmation of findings; Self-correction; Control of unwanted factors.

What is a Research Design?

A general plan for conducting research specifying sampling; variables; measurements; and statistical procedures.

Name and define the three main research designs

Cross-sectional: snapshot at one time; Longitudinal: repeated measures over time; Experimental: highest control over extraneous variables.

Why do we sample instead of conducting a census?

Practical reasons: budget; time constraints; limited access; still provides reliable conclusions if well-designed.

Define Population

Any complete group of entities sharing a common characteristic.

Define Sample

A subset of a population selected for study.

Define Sampling Frame

A list of the target population from which the sample is drawn.

Name two principles of good sampling

Precision (adequate sample size) and Representativeness (sample reflects population characteristics).

Differentiate between parameter and statistic

Parameter: describes population (μ σ²) Statistic: describes sample (x̄ s²).

What are the two main branches of statistics?

Descriptive (summarize data) and Inferential (test hypotheses and generalize to population).

Define Nominal scale

Categories with no order (e.g. gender or department).

Define Ordinal scale

Categories with an order or rank but unequal intervals (e.g. satisfaction ratings).

Define Interval scale

Ordered scale with equal intervals but no true zero (e.g. temperature or calendar dates).

Define Ratio scale

Ordered scale with equal intervals and a true zero (e.g. income or weight).

What is a Likert scale?

A scale used to measure attitudes or opinions with ordered agreement levels (e.g. 1=strongly disagree to 5=strongly agree).

What is a Semantic Differential scale?

A scale with two labeled opposite adjectives (e.g. Clear — Unclear) with a continuum between them.

Define Mean

Arithmetic average: sum of all scores ÷ number of scores.

Define Median

The middle value when data are ordered; divides data into two equal halves.

Define Mode

The most frequently occurring score in a distribution.

When is the median preferred over the mean?

When the data are skewed or contain outliers.

Define Range

Difference between the largest and smallest observation (max - min).

Define Variance

The average of squared deviations from the mean; measures total spread of data.

Sample Variance formula

s² = Σ(x – x̄)² ÷ (n – 1)

Sample Standard Deviation formula

s = √s² (square root of sample variance).

Define Interquartile Range (IQR)

Difference between the third quartile (Q3) and first quartile (Q1); shows spread of middle 50% of data.

What does a positively skewed distribution look like?

Tail is longer on the right; cluster of low scores on the left.

What does a negatively skewed distribution look like?

Tail is longer on the left; cluster of high scores on the right.

Define Unimodal distribution

Distribution with one peak or mode.

Define Bimodal distribution

Distribution with two prominent peaks.

Define Multimodal distribution

Distribution with more than two peaks.

What is a Histogram?

A bar graph showing frequency distribution for quantitative data.

What is a Frequency Polygon?

A line graph connecting the midpoints of histogram bars to show distribution shape.

What is a Boxplot?

A graphical summary using median quartiles and whiskers to visualize distribution and detect outliers.

List common Sampling Errors

Sampling frame error; Random sampling error; Nonresponse error.

Probability vs. Non-Probability Sampling

Probability: random known chance of selection. Non-probability: subjective unknown chance of selection.

What is a Census?

An investigation including the entire population (population = sample).

What is a normal distribution?

A symmetrical bell-shaped distribution where mean median and mode are equal and most values cluster around the center.

What are the properties of the normal curve?

Symmetrical shape; mean = median = mode; predictable proportions of scores within standard deviations.

What does Chebyshev’s theorem or the Empirical Rule state?

For normal data about 68 percent of values lie within 1 SD of the mean; 95 percent within 2 SD; 99 percent within 3 SD.

Define a Z-score

The number of standard deviations a data point is from the mean.

Z-score formula

(X – M) ÷ SD

What does a positive Z-score mean?

The score is above the mean.

What does a negative Z-score mean?

The score is below the mean.

What are Confidence Intervals (CI)?

Sample-based estimates of a range likely to include the population characteristic with a given probability.

What does a 95 percent confidence interval mean?

We are 95 percent confident that the true population mean lies within the calculated interval.

What Z-values correspond to a 95 percent confidence interval?

-1.96 and +1.96

What Z-values correspond to a 99 percent confidence interval?

-2.57 and +2.57

General formula for CI of the mean

Mean ± Z * (Standard error)

Define Range

The difference between the maximum and minimum values in a dataset.

Limitation of Range

Only considers two values and ignores the rest of the data.

Define Mean Absolute Deviation (MAD)

The average of absolute differences between each observation and the mean.

Define Variance

The average of squared deviations from the mean; measure of total spread.

Sample Variance formula

s² = Σ(x – x̄)² ÷ (n – 1)

Define Standard Deviation

The square root of variance; shows average spread of scores around the mean.

What does a small SD indicate?

Low variability; data points close to the mean.

What does a large SD indicate?

High variability; data points widely spread from the mean.

Define Correlation

Statistical measure of the strength and direction of the relationship between two variables.

Pearson correlation coefficient (r)

Used when both variables are interval or ratio; measures linear relationship.

Range of r

From -1 to +1

Interpret r = +1

Perfect positive correlation.

Interpret r = -1

Perfect negative correlation.

Interpret r = 0

No linear relationship.

When should Pearson correlation be used?

When both variables are interval or ratio and assumptions of linearity and normality are met.

When should Spearman correlation be used?

When one or both variables are ordinal or ranked; or when data are not normally distributed.

What is a scatterplot?

Graph plotting paired data to visualize relationships between two variables.

Define Coefficient of Determination (r²)

The proportion of variance in the dependent variable explained by the independent variable.

If r = 0.7 what is r²?

0.49; 49 percent of variation explained by the independent variable.

What is Effect Size?

The magnitude of a relationship; for correlation calculated as r².　(coefficient of determination)

Correlation vs Causation

Correlation means association but does not imply causation.

What is a Crosstab?

A table showing frequency distribution between two categorical variables.

What is Chi-Square test used for?

To test whether two categorical variables are independent or related.

Null hypothesis in Chi-Square test

Both variables are independent.

When is Chi-Square significant?

When observed frequencies differ greatly from expected frequencies suggesting variables are related.

When should correlation not be used?

When the relationship is non-linear or when variables are categorical.

What is the purpose of inferential statistics?

To make conclusions about populations from samples using probability.

What is hypothesis testing?

A statistical procedure to decide whether an observed effect is real or could have occurred by chance.

What is a null hypothesis (H0)?

The assumption of no relationship; no difference; no effect; usually includes equality (=; ≥; ≤).

What is an alternative hypothesis (H1)?

The assumption of a relationship; difference; or effect; does not include equality.

Why must H0 and H1 come as a pair?

Because they are complements; one assumes no effect and the other assumes an effect.

What does the phrase "If p is low; H0 must go" mean?

If the p-value is less than the significance level (typically 0.05); we reject the null hypothesis.

What does inability to reject H0 mean?

It does not prove H0 is true; it only means there is not enough evidence to reject it.

What is a directional hypothesis?

A one-tailed test where we expect a specific direction of effect (e.g. greater than or less than).

What is an exploratory hypothesis?

A two-tailed test where we only expect a difference but not a specific direction.

What does a p-value represent?

The probability of obtaining results at least as extreme as the observed ones if H0 is true.

What is the typical threshold for significance?

p-value less than 0.05.

What does marginal significance mean?

p-value between 0.05 and 0.10; result is almost significant but not quite.

What is Type I error?

Rejecting H0 when it is actually true; falsely finding an effect.

What is Type II error?

Failing to reject H0 when it is actually false; missing a real effect.

Which is considered more serious: Type I or Type II error?

Type I error is generally considered more serious.

How is the probability of Type I error controlled?

By setting the significance level alpha (e.g. 0.05 or 0.01).

How can Type II errors be reduced?

By increasing sample size or reducing variability between groups.

What role does probability play in hypothesis testing?

It benchmarks chance; results with less than 5 percent probability under H0 are considered rare and significant.

What does the James Bond martini example illustrate?

Testing H0 (Bond cannot tell the difference) vs H1 (Bond can); calculating probability of his correct guesses under H0.

What is a one-sample t-test?

Test to compare a sample mean against a population mean or external norm.

What is an independent-samples t-test?

Test to compare means of two independent groups.

What is a paired-samples t-test?

Test to compare means of two related groups or the same group under two conditions.