Stat for Psych

Population

A large group of elements about which we wish to make an inference

Parameter

A number that summarizes a characteristic of the population

Sample

A subset of population selected for analysis

Statistic

A number that summarizes some characteristic of the sample

Variable

A dimension on which quantifiable change exits

Ex: Biological sex, mood, race, and income

Nominal Scale

Levels are descriptive nonnumeric categories without an ordered relationship

Ex: biological sex and race

Ordinal Scale

Levels are numeric quantities with a specific order but no clear uniform spacing between them

Interval Scale

Levels are numeric quantities with a specific order and equal spacing

Ratio Scale

An interval scale with a true 0 point

Discrete Variable

Has a finite or countably infinite number of values

Ex: Number of symptoms and number of smiles per week

Continuous Variable

Infinite number of possible values between each value

Frequency Table

Displays how frequently each level of a variable appeared in your sample

Ex: How many males there were, how many 85's there were on a test

Cumulative Frequency

The sum of the frequencies for that value or lower

Percentile Rank

The sum of the percentages of that value and lower

Grouped Frequency Table

Same thing as a frequency table but for intervals, so how often that interval appeared

Bar Graph

Frequency distribution which is plotted horizontally with spaces between them, usually used for nominal scales

Histogram

Values or intervals of a numeric value (usually continuous), are plotted horizontally, basically a bar graph without spacing

Xᵢ

An individual score of variable "X"

µᵪ

Population mean of variable "X"

X̅

Sample mean of variable "X"

∑

Sum of values across a set of values

Npop

Population Size

N

Sample Size

Central Tendency

A representative, or typical value. A single number that represents the general location of a set of scores.

Mean

Indicator of Central Tendency, the average

Median

Indicator of Central Tendency, the middle value

Mode

Indicator of Central Tendency, the most commonly occurring variable

Formula for population Mean

μᵪ = ∑Xᵢ / Npop

Formula for sample Mean

X̅ = ∑Xᵢ / N

Positive Skewness

A small number of extreme highs pull the mean above the median

Negative Skewness

A small number of extreme lows pulls the mean below the median

Variability

How spread out the scores are around the mean

Range

Indicator of variability, difference between the highest and lowest values

Variance

The average squared deviation from the mean

σ²ᵪ

Population variance

S²ᵪ

Sample Variance

Standard Deviation

The average difference between a score and the mean

σᵪ

Population Standard Deviation

Sᵪ

Sample Standard Deviation

Deviation score

Xᵢ - μᵪ

Squared Deviation score

(Xᵢ - μᵪ )²

Sum of the Squared Deviations

∑(Xᵢ - μᵪ )²

Variance Formula

Variance ∑(Xᵢ - μᵪ )²/Npop

Standard Deviation

√(∑(Xᵢ - μᵪ )²/Npop)

Sample Based Estimate of Population Variance

∑(Xᵢ - X̅)²/N-1

Sample Based Estimate of Population Standard Deviation

√(∑(Xᵢ - X̅)²/N-1)

Frequency Distribution

The pattern of frequencies across the various levels of a variable

Unimodal Distribution

One value, or one interval of values, is higher

Bimodal Distribution

Two values, or intervals of values, are higher than the others and are at similar heights

Symmetrical Distribution

Same number of scores above and below the mean, and the right and left sides are mirror images

Skewed Distribution

A distribution that isn't symmetrical where more scores are on one side and other side contains extreme scores

Normal Distribution

Unimodal, symmetrical, and mesokurtic (bell-shaped)

Z Score (Standard Score)

A type of transformed of a numeric variable expressed in terms of a distribution that has a mean of 0 and a SD of 1, the unit of measurement is the SD, positive scores are above average and negative are below average

Population Z-Score Formula

zXᵢ = (Xᵢ - μᵪ)/σᵪ

Sample Z-Score Formula

zXᵢ = (Xᵢ - X̅)/Sᵪ

Frequentist Interpretation of Probability

The expected relative frequency of a target outcome

Bayesian Interpretation of Probability

An estimate of the likelihood of a target outcome

Central Limit Theorem

If each observation in a set of observations is determined by a large number of random factors, the distribution will likely approach a normal distribution

Percentage of Observations that falls within 1 SD from the Mean in a Normal Curve

68.2%

Percentage of Observations that falls between 1 and 2 SDs from the Mean in a Normal Curve

27.2%

Percentage of Observations that falls outside of 2 SDs from the Mean in a Normal Curve

4.6%

In a normal curve where do 95% of scores fall?

1.96 SDs

Standard Normal Curve

A normal curve with a mean of zero and a SD of 1

Sampling Distribution of the Mean, or S.D.M., μx̅

The distribution of means from repeated samples of size N drawn from a population, always converges to the Population Mean

μx̅ = μx

The mean of the S.D.M. equals the population mean

σx̅

standard deviation of the sampling distribution of x̅, A.K.A. the Standard Error (SE)

σx̅ < σx

The variability of the distribution of sample means is less than the population of variability on x

Standard Error Formula

σx̅ = σx/√N

Population Distribution, Sample Distribution, and S.D.M.

1. μᵪ = ∑Xᵢ / Npop -> √(∑(Xᵢ - μᵪ )²/Npop)

2. X̅ = ∑Xᵢ / N -> √(∑(Xᵢ - X̅)²/N-1)

3. μx̅ = μx -> σx̅ = σx/√N

Null Hypothesis

Statement that one's research hypothesis is not true, if the null is rejected you're hypothesis is upheld

P-value

Probability that if the null is true, one would have obtained a sample statistic as or more extreme as what one obtained in one's sample

Z-Test

Procedure for testing hypothesis that a target population from which you've drawn a sample differs from another population with a known mean and variability, assuming both populations have the same variance

Cutoff for rejecting the null

+-1.96

Zstat formula

(X̅-μx̅)/σx̅

T-test

Statistical test comparing means between two groups, used when the population standard deviation is unknown, to calculate this we need to find the estimated sample error, you test the null

Estimated sample error formula

SX̅ = Sx/√N

Degrees of freedom

N-1, represents the number of independent values in a dataset that are free to vary when estimating statistical parameters

Tstat formula

(X̅-μx̅)/SX̅

DDM, Distribution of Difference Between Means

Distribution of differences between the means of pairs of samples such that, for each pair of samples, one is from one population and the other is from another population

Estimated Standard Error of the DDM

The estimated standard deviation of the sampling distribution for the difference between two independent sample means

Distribution of Difference Between Means Formula

(X1-X2)/SX1-X2

Confidence Interval

A range of values, derived from sample data, that is likely to contain the value of an unknown population parameter

Confidence Interval Formula

(X̅1-X̅2) +/- (SX̅1-X̅2 * Tcrit)

T-test for matched samples

Looking for the difference score where each variable is 2 samples

D

Difference score

Formula

td̅ = (D̅-µD̅)/SD̅ = D̅/SD̅

SD̅

Standard Deviation of Difference Score

SD̅ formula

SD/√N

Confidence Interval for difference score

D̅ + / - (SD̅ * tcrit)

Margin of error

(XXX * tcrit)

Quasi-Independent Variable

A variable not controlled by the experimenter. Instead participants already belong to the groups being studied. It is also called a subject variable, selected variable, or grouping variable. Ex: Gender

Deciles

Deciles divide a distribution into 10 equal parts. Each part represents 10% of the data. The 9th decile marks the point below which 90% of the scores fall, so the top 10% lies above the 9th decile.

Quartiles

Quartiles divide a distribution into 4 equal parts. Each part represents 25% of the data. The 2nd quartile is the 50th percentile, which is also the median.

Frequency Polgyon

A graph similar to a histogram. Instead of bars you have plot points representing frequencies at each score or class interval and then connect the points with straight lines, it is useful for showing the overall shape of a distribution

Stem-and-leaf display

A Stem and leaf display organizes data in a way that combines features of a frequency distribution and a histogram. The stems represent the leading digits or class intervals, and the leaves represent the individual values within each stem. It is useful because it shows the shape of the distribution while still keeping the original data values visible.

Weighted Mean

A weighted mean, or weighted average, is a mean in which some values count more than others. Each score is multiplied by it’s weight, usually it’s frequency or importance. Example: GPA. Letter grades are assigned numerical values, and each grade is weighed by the number of credit hours or courses receiving that grade.

Separate-Variance t Test

A separate-variance t test compares the means of two independent groups when the assumption of equal variances is not met. It is especially useful when the groups have different variances and/or different sample sizes. This test produces a more accurate p-value and confidence interval and helps reduce inflated Type 1 error rate.

Matched-Pairs Design

A matched-pairs design is an experimental design in which participants are placed into pairs based on similar characteristics, such as age, gender, or pretest score. One member of each pair receives the treatment, and the other receives a different measure or control. This design helps reduce confounding variables, increase precision, and allow for smaller sample sizes.

Point estimate

single numeric value that is your best estimate of the population parameter

Linear Correlation

Association between 2 variables in which dots on a scatter diagram roughly follow a straight line