Reasoned for 12 seconds

Below is an expanded and structured guide to the most common statistical formulas, complete
with in‐depth explanations and additional metrics (like effect sizes). Use this as a handy reference
for a wide range of statistical techniques.

1. Basic Summation Notation

When dealing with a sample of n data points x1 , x2 , … , xn (and possibly a corresponding set
​ ​ ​

y1 , y2 , … , yn ), the following summation symbols are used frequently:

​ ​ ​

1. Sum of values:
n
∑ xi ​ ​ (Add up all xi ). ​

i=1

2. Sum of squares:
n
∑ x2i ​ ​ (Add up the squares of each xi ). ​

i=1

3. Sum of products (for paired data xi , yi ): ​ ​

n
∑ xi y i​ ​ ​ (Add up products xi ⋅ yi ). ​ ​

i=1

Concept: These summations form the building blocks in the formulas for variance, correlation, and
regression.

2. Descriptive Statistics: Mean and Variance

2.1 Sample Mean

For a set of n observations x1 , x2 , … , xn , the sample mean is:
​ ​ ​

n
1
ˉ =
x ∑ xi . ​ ​ ​

n i=1

Interpretation: This is the “average” value in the sample.

2.2 Sample Variance

The sample variance s2 (using Bessel’s correction) is:

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 n
1
∑(xi − x
2 2
s = ˉ) .
n−1
​ ​ ​

i=1

Interpretation: Measures the average squared deviation from the mean, using n − 1 in the
denominator for an unbiased estimate.

2.3 Sample Standard Deviation

s = s2 . ​

Interpretation: The typical (average) distance of points from the mean, in the same units as x.

3. Z‐Test (Large Sample or Known Population Variance)

Sometimes we use a z‐test instead of a t‐test when:

1. The population variance σ 2 is known, or

ˉ is approximately normally
2. The sample size n is large (and by the Central Limit Theorem, x
distributed).

3.1 One‐Sample z‐Test

Hypotheses (two‐sided):

H0 : μ = μ 0 ,
​ ​ H1 : μ =
 μ0 .
​ ​

Test Statistic:

ˉ − μ0
x
z = .
​

σ/ n ​

Concept: A z‐test is like a t‐test but uses the population standard deviation σ directly (or uses a
large‐sample approximation).

4. T‐Tests

4.1 One‐Sample t‐Test

Used to test whether the sample mean differs from a hypothesized value μ0 . ​

Hypotheses:

H0 : μ = μ 0 ,
​ ​ H1 : μ =
 μ0 .
​ ​

Test Statistic:

ˉ
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 ˉ − μ0
x
t = ,
​

s/ n ​

with df = n − 1.

Concept: Compares the observed mean to the hypothesized mean in units of the estimated
standard error s/ n. ​

4.2 Paired t‐Test

Use when two measurements are on the same subjects (e.g., “before” vs. “after”).

1. Compute differences: di = xi,before

​ ​ − xi,after . ​

2. Treat the di as a single sample.

​

Hypotheses:

H0 : μd = 0,
​ H1 : μ d 
= 0.
​ ​

Test Statistic:

dˉ
t = ,
sd / n
​

​ ​

where dˉ and sd are the mean and standard deviation of the differences.
​

Concept: Controls for “within‐subject” variability, which often increases power compared to an
independent‐samples test.

4.3 Two‐Sample t‐Test

Compares means from two independent groups, assuming (near) normality.

4.3.1 Equal Variance (Pooled) t‐Test

Pooled variance:

(n1 − 1) s21 + (n2 − 1) s22

s2p = .
​ ​ ​ ​

n1 + n2 − 2
​ ​

​ ​

Test Statistic:

ˉ1 − x
(x ˉ2 ) − (μ1 − μ2 )0
t = ,
​ ​ ​ ​ ​

1 1
sp ​

n1 ​
​ + n2 ​
​ ​

often (μ1 − μ 2 )0 = 0 .
​ ​

4.3.2 Unequal Variance (Welch’s) t‐Test

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 Test Statistic:
ˉ1 − x
x ˉ2
t = .
​ ​

s21 s22
+
​ ​

n1 n2
​ ​ ​

​ ​

Degrees of freedom: Approx. via Welch–Satterthwaite formula.

Concept: The two‐sample t‐test (pooled or Welch) checks if two population means differ, allowing
for either equal or unequal variances.

5. One‐Way ANOVA
Used to compare more than two group means, under an assumption of normality and equal
variances.

Hypotheses:

H0 : μ 1 = μ 2 = ⋯ = μ a ,
​ ​ ​ ​ H1 : at least one μj differs.
​ ​

5.1 Sums of Squares

1. Between‐Groups sum of squares (SSB):
a
SSB = ∑ nj (x
2
ˉj − x
ˉoverall ) .
​ ​ ​ ​

j=1

2. Within‐Groups sum of squares (SSW):

a nj ​

SSW = ∑ ∑(xij − x
2
ˉj ) . ​ ​ ​ ​

j=1 i=1

3. Total sum of squares (SST):

SST = SSB + SSW.

5.2 Mean Squares and F‐Statistic

SSB SSW
MSB = , MSW = .
a−1 ∑(nj − 1)
​

MSB
F = , df = (a − 1, ∑(nj − 1)).
MSW
​ ​

Concept: Checks if the variation between group means is significantly larger than random within‐
group variation.

6. Two‐Way ANOVA (Without Replication)

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Used to analyze effects of two categorical factors (e.g., Factor A and Factor B), each with multiple
levels, but with exactly one observation per cell (no replication).

Sums of Squares:
SSA: Variation due to factor A (rows).
SSB: Variation due to factor B (columns).
SSE: Residual (error) = SST − SSA − SSB .
Mean Squares:

SSA SSB SSE

MSA = , MSB = , MSE = .
a−1 b−1 (a − 1)(b − 1)
​ ​ ​

F ‐Statistics for each factor:

MSA MSB
FA = , FB = .
MSE MSE
​ ​ ​ ​

Concept: Separates out the effect of each factor while controlling for the other.

7. Chi‐Square (χ2 ) Test of Independence

Used for categorical data in an r × c contingency table of counts.

1. Compute row totals, column totals, grand total.

2. Expected frequency if the row and column variables are independent:

(row i total) × (column j total)

Eij = .
grand total
​ ​

3. Chi‐Square Statistic:

(Oij − Eij )2
χ = ∑2
.
​ ​

​ ​

Eij
all cells
​

4. Degrees of freedom: (r − 1) × (c − 1).

Concept: Checks whether the pattern of counts deviates from what we’d expect if the two
categorical variables were truly independent.

8. Nonparametric (Rank) Tests

8.1 Wilcoxon Rank‐Sum (Mann–Whitney) Test

For two independent samples when we cannot assume normality.

1. Combine the two groups into one set, rank from smallest to largest.
2. Let W = sum of ranks for one group.
3. Compare W to the critical distribution (or compute a p‐value).
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Concept: Uses ranks instead of raw values, making it more robust to non‐normal data.

9. Correlation Coefficient (Pearson’s r)

Measures linear association between X and Y .

n ∑ xi yi − (∑ xi )(∑ yi )
r = .
​ ​ ​ ​

[n ∑ x2i − (∑ xi )2 ] [n ∑ yi2 − (∑ yi )2 ]
​ ​ ​ ​ ​

Range: −1 ≤ r ≤ 1.
r ≈ +1 → strong positive association; r ≈ −1 → strong negative; r ≈ 0 → weak linear
relationship.

Concept: Standardized measure of the direction and strength of a linear relationship.

10. Simple Linear Regression

Fits a line Y = a + bX to predict Y from X .

1. Slope b:

n ∑ XY − (∑ X)(∑ Y )
b = .
n ∑ X 2 − (∑ X)2
​

2. Intercept a:

∑ Y − b (∑ X)
a = . ​

n
3. Predicted value at X = x:

Y^ = a + b x.

10.1 Coefficient of Determination (R2 )

A measure of goodness‐of‐fit:

SSR
R2 = where SSR = ∑(Y^i − Yˉ )2 , SST = ∑(Yi − Yˉ )2 .
SST
​ ​ ​

Interpretation: R2 is the proportion of the variance in Y explained by the linear model with
X.

11. Standard Errors and Confidence Intervals

11.1 Standard Error of the Mean

s
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 s
SE(x
ˉ) = . ​

n ​

Interpretation: How far the sample mean will typically be from the true mean (the smaller, the
more precise).

11.2 Confidence Interval for μ

(assuming the sample is from a normal population or n is large)

s
ˉ ± tα/2, n−1 ×
x ​
. ​

n ​

Interpretation: A range of plausible values for μ. If the same procedure is repeated many
times, ~95% (for α = 0.05) of those intervals would contain the true mean.

12. Additional Effect Size Formulas

Effect sizes supplement p‐values by indicating practical (rather than just statistical) significance.

12.1 Cohen’s d (for t‐Tests)

One‐sample or Paired t‐test:

ˉ − μ0
x dˉ
d = or .
​

​ ​

s sd​

Two‐sample t‐test:

ˉ1 − x
x ˉ2
d = ,
​ ​

sp ​

where sp is the pooled standard deviation.

​

Interpretation (rough guideline):

d ≈ 0.2 = small effect,

d ≈ 0.5 = medium,
d ≈ 0.8 = large.

12.2 Eta‐Squared (η 2 ) in ANOVA

One‐Way ANOVA:

SSB
η2 = .
SST
​

Partial Eta‐Squared in more complex designs (e.g., Two‐Way):

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SS 7/16
 2 SSeffect
= .
​

ηpartial
SSeffect + SSerror
​ ​

​ ​

Interpretation: η 2 is the proportion of total variation attributable to a factor.

Concluding Notes
1. Check Assumptions:
Normality for z/t‐tests and ANOVA (or at least approximate normality).
Equal variances for certain t‐tests (pooled) or standard one‐way ANOVA (though it can be
robust).
Random/independent samples whenever required.
2. Tables or Software:
Critical values (tα,df , Fα,df1 ,df2 , χ2α,df ) are found in statistical tables or software output.
​

​ ​
​ ​

For large samples, p‐values come directly from software.

3. Effect Sizes:
Go beyond p‐values to understand the magnitude of differences or relationships.
4. Nonparametric Methods:
Use when normality or other assumptions are violated. They rank data rather than

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Reasoned for 5 seconds

Below is an expanded, step‐by‐step explanation of the key formulas you might encounter in basic
statistical analyses. For each formula, we break down:

1. What the formula is used for (the “why”).

2. Each term’s meaning (the “what”).
3. Interpretation (the “how to understand” or “what it tells us”).

Feel free to scan for the particular test or concept you need.

1. Summation Notation
n
You will often see expressions like ∑i=1 xi . This is shorthand for “sum xi over i from 1 to n.”
​ ​ ​

n
∑i=1 xi means x1 + x2 + ⋯ + xn .
​ ​ ​ ​ ​

n
∑i=1 x2i means x21 + x22 + ⋯ + x2n .
​ ​ ​ ​ ​

n
∑i=1 xi yi means x1 y1 + x2 y2 + ⋯ + xn yn .
​ ​ ​ ​ ​ ​ ​ ​ ​

Why it matters: These summations are building blocks for almost every other formula in statistics.

2. Descriptive Statistics

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 ˉ
2.1 Sample Mean x
n
1
ˉ =
x ∑ xi .
​ ​ ​

n
i=1

xi : The ith observation in your sample.

​

n: The total number of observations.

n
∑i=1 xi : The sum of all observations.
​ ​

ˉ (“x‐bar”): The arithmetic mean of the sample.

x

ˉ is the “average value” and serves as a measure of central tendency.

Interpretation: x

2.2 Sample Variance s2

n
1
2
s = ∑(xi − x
ˉ )2 .
n − 1 i=1
​ ​ ​

xi : The ith observation.

​

ˉ: The sample mean.

x
(xi − x ˉ): The deviation of the ith observation from the mean.
​

(xi − x ˉ)2 : Squared deviation (removes negative signs).

​

n − 1: Called Bessel’s correction; used to make s2 an unbiased estimator of the true

(population) variance.
s2 : The average of the squared deviations (with the slight correction in the denominator).

Interpretation: Variance represents how spread out your data are. A larger s2 means more spread.

2.3 Sample Standard Deviation s

s= s2 . ​

s2 : Sample variance (from above).

s: Standard deviation in the original units of x.

Interpretation: s is the average distance of data points from the sample mean (roughly).

3. Z‐Test (When Population Variance is Known or n is Large)

ˉ − μ0
x
z= .
​

σ/ n ​

ˉ: Sample mean.
x
μ0 : Hypothesized population mean (under H0 ).
​ ​

σ : Known (or assumed known) population standard deviation.

n: Sample size.

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 σ/ n: The standard error of the mean when σ is known.
​

ˉ is from μ0 . Compare z to a standard

Interpretation: Tells you how many “standard errors” away x ​

normal table (or get a p‐value) to test H0 ​ : μ = μ0 . ​

4. T‐Tests
4.1 One‐Sample t‐Test
ˉ − μ0
x
t= .
​

s/ n ​

ˉ: Sample mean.
x
μ0 : Hypothesized population mean under the null H0 .
​ ​

s: Sample standard deviation.

n: Square root of the sample size.
​

s/ n: Estimated standard error of the mean.

​

t: The resulting test statistic, with (n − 1) degrees of freedom.

Interpretation: If ∣t∣ is large, it suggests x

ˉ differs significantly from μ0 . Compare to the t ​

distribution with n − 1 degrees of freedom.

4.2 Paired t‐Test

1. Compute differences: di = xi,before

​ ​ − xi,after . ​

2. Then, treat di as a single sample:

​

dˉ
t= .
sd / n
​

​ ​

di : The difference for the ith subject (between two paired measurements).
​

dˉ: The mean of all differences.

sd : The standard deviation of the differences.
​

n: The number of pairs.

sd / n: Standard error of the mean difference.
​ ​

Interpretation: Measures whether the average difference is significantly different from zero,
controlling for within‐subject variability.

4.3 Two‐Sample t‐Test

Compare two independent groups.

4.3.1 Equal Variance (Pooled) t‐Test

1. Pooled Variance:

2 2
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 (n1 − 1)s21 + (n2 − 1)s22
s2p = .
​ ​ ​ ​

n1 + n2 − 2
​ ​

​ ​

n1 , n2 : Sample sizes in group 1 and group 2.

​ ​

s21 , s22 : Variances in group 1 and group 2.

​ ​

s2p : Weighted average of the two group variances (assumes both groups truly have the
​

same variance in the population).

2. Test Statistic:

(x
ˉ1 − x
ˉ2 ) − (μ1 − μ2 )0
t= .
​ ​ ​ ​ ​

1 1
sp ​

n1 ​
​ + n2 ​
​ ​

ˉ1 , x
x ˉ2 : Sample means in group 1 and group 2.
​ ​

(μ1 − μ2 )0 : The hypothesized difference in population means (often 0).

​ ​ ​

sp : Square root of the pooled variance.

​

1 1
n1 + n2 : Summation inside a square root for the standard error of the difference in
​ ​

​ ​

means.

Interpretation: If ∣t∣ is large, it suggests the two groups have different means. Degrees of freedom
= n1 ​ + n2 − 2. ​

4.3.2 Welch’s t‐Test (Unequal Variances)

ˉ1 − x
x ˉ2
t= .
​ ​

s21 s22
+
​ ​

n1 n2
​ ​ ​

​ ​

s21 , s22 : Sample variances of group 1 and group 2 (not pooled because we assume they differ).
​ ​

n1 , n2 : Sample sizes.
​ ​

The denominator: Standard error for difference in means under unequal variances.
Degrees of freedom: Calculated via the Welch–Satterthwaite approximation (not simply n1 ​
+
n2 − 2).
​

Interpretation: More flexible—no assumption of equal group variances.

5. One‐Way ANOVA (Analysis of Variance)

Used to compare more than two groups.

Hypothesis:

H0 : μ 1 = μ 2 = ⋯ = μ a ,
​ ​ ​ ​ H1 : At least one μj 
​ = others. ​

5.1 Sums of Squares

1. Between‐Groups (SSB):

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 a
SSB = ∑ nj (x
2
ˉj − x
ˉoverall ) .
​ ​ ​ ​

j=1

a: Number of groups.
nj : Sample size in group j .
​

ˉj : Mean of group j .
x ​

ˉoverall : Grand mean (overall average).

x ​

Interpretation: The variation between group means (how far each group mean is from
the overall mean).
2. Within‐Groups (SSW):
a nj ​

SSW = ∑ ∑(xij − x
2
ˉj ) .
​ ​ ​ ​

j=1 i=1

xij : The ith observation in group j .

​

ˉj : Mean of group j .
x ​

Interpretation: The variation within each group (how spread out the data are inside each
group).
3. Total (SST):

SST = SSB + SSW.

5.2 Mean Squares and F‐Statistic

SSB SSW
MSB = , MSW = .
a−1 ∑(nj − 1)
​

MSB
F = with df = ( a − 1, ∑(nj − 1) ).
MSW
​ ​

MSB: “Mean Square Between” = average between‐group variance.

MSW: “Mean Square Within” = average within‐group variance.
F : If the between‐group variance is large relative to the within‐group variance, F is large ⇒
evidence that at least one group mean differs significantly.

Interpretation: A large F means that the differences among group means likely exceed what we’d
expect by random chance.

6. Two‐Way ANOVA (Without Replication)

Examines effects of two categorical factors (e.g., Factor A with a levels, Factor B with b levels) when
there’s only one observation per combination of levels (no replication).

SSA: Variation due to Factor A (rows).

SSB: Variation due to Factor B (columns).
SSE: Error = SST − SSA − SSB .

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Then:

SSA SSB SSE

MSA = , MSB = , MSE = .
a−1 b−1 (a − 1)(b − 1)
​ ​ ​

MSA MSB
FA = , FB = .
MSE MSE
​ ​ ​ ​

Interpretation: Tests if Factor A levels differ on average, and if Factor B levels differ on average,
controlling for each other’s effect.

7. Chi‐Square (χ2 ) Test of Independence

Used for categorical data in an r × c table (contingency table) of counts.

1. Observed counts: Oij in row i, column j .

​

2. Expected counts (if independent):

(row i total) × (column j total)

Eij = .
grand total
​ ​

3. Chi‐Square Statistic:

(Oij − Eij )2
χ2 = ∑ .
​ ​

​ ​

Eij
all cells
​

4. Degrees of freedom: (r − 1)(c − 1).

Interpretation: If χ2 is large, there is evidence that row and column variables are not independent.

8. Nonparametric (Rank) Tests

8.1 Wilcoxon Rank‐Sum (Mann–Whitney) Test

For two independent samples when normality is questionable.

1. Combine all observations from both groups, rank them from smallest to largest.
2. Let W = sum of ranks for one group (or the smaller group).
3. Use special tables or software to determine significance.

Interpretation: Tests whether the distribution of one group tends to have larger or smaller values
than the other group, without assuming normality.

9. Correlation Coefficient (Pearson’s r )

( to PDF
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 n ∑ xi y i − ( ∑ xi ) ( ∑ y i )
r= .
​ ​ ​ ​

[n ∑ x2i − (∑ xi ) ] [n ∑ yi2 − (∑ yi ) ]
2 2
​ ​ ​ ​ ​

xi , yi : Paired data points.

​ ​

n ∑ xi yi − (∑ xi )(∑ yi ): Numerator that captures the co‐variation of x and y .

​ ​ ​ ​

Denominator: Product of the square roots of each variable’s sum‐of‐squares term, ensuring r
is normalized between –1 and +1.
r: The Pearson correlation.

Interpretation:

r ≈ +1: Strong positive (as x increases, y increases).

r ≈ −1: Strong negative (as x increases, y decreases).
r ≈ 0: Little linear relationship.

10. Simple Linear Regression (Least Squares)

Fits a line Y = a + bX to predict Y from X .

1. Slope b:

n ∑ XY − (∑ X)(∑ Y )
b= .
n ∑ X 2 − (∑ X)2
​

n: Number of (X, Y ) pairs.

∑ X, ∑ Y : Sum of X values, sum of Y values.
∑ X 2 : Sum of squares of X .
∑ XY : Sum of products.
2. Intercept a:

∑ Y − b (∑ X)
a= . ​

n
^ at X
3. Predicted value Y = x:

Y^ = a + b x.

10.1 Coefficient of Determination (R2 )

SSR
where SSR = ∑(Y^i − Yˉ ) , SST = ∑(Yi − Yˉ ) .
2 2
R2 =
SST
​ ​

Yˉ : Mean of all observed Yi . ​

Y^i : Predicted Y value from the model for the ith observation.
​

SSR (Regression Sum of Squares): Variation explained by the regression model.

SST (Total Sum of Squares): Total variation in Y .

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Interpretation: R2 is the proportion of variance in Y explained by X . Closer to 1 ⇒ better fit.

11. Standard Error and Confidence Intervals

11.1 Standard Error of the Mean (SEM)
s
SE(x
ˉ) = . ​

n ​

s: Sample standard deviation.

n: Sample size.

ˉ might deviate from the true population

Interpretation: The typical distance the sample mean x
mean μ.

11.2 Confidence Interval for μ

For a normal population (or large n):

s
ˉ ± tα/2, n−1 ×
x ​ . ​

n ​

tα/2, n−1 : Critical t‐value from the t distribution for a chosen confidence level (e.g., 95%).
​

α: Significance level (e.g., 0.05).

ˉ ± (… ): Lower and upper boundaries of the CI.
x

Interpretation: Gives a range of plausible values for μ. If repeated many times, ~95% of such
intervals would contain the true μ (for 95% confidence).

12. Effect Size Measures

12.1 Cohen’s d

One‐sample/Paired:

ˉ − μ0
x dˉ
d= or .
​

​ ​

s sd​

Two‐sample:

ˉ1 − x
x ˉ2
d= .
​ ​

sp ​

ˉ1 , x
x ˉ2 : Means of the two groups.
​ ​

sp : Pooled standard deviation.

​

Interpretation (common guidelines):

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 d ≈ 0.2 = small,
d ≈ 0.5 = medium,
d ≈ 0.8 = large.

12.2 Eta‐Squared (η 2 ) in ANOVA

SSB
η2 = .
SST
​

SSB: Between‐group sum of squares.

SST: Total sum of squares.

Interpretation: Fraction of total variation explained by the factor (like “proportion of variance
accounted for”).

Concluding Remarks
1. Know Your Assumptions
Normality: z‐tests, t‐tests, standard ANOVA typically assume normal distributions (or at
least approximate).
Equal variances: Some t‐tests (the pooled version) and standard one‐way ANOVA assume
equal variances across groups (though they’re somewhat robust if group sizes are
similar).
Independence: Observations typically must be independent unless you specifically use
paired or repeated‐measures designs.
2. Degrees of Freedom (df)
Always keep track of df for each test (e.g., n − 1, n1 ​ + n2 − 2, or (r − 1)(c − 1)).
​

They dictate which distribution to use for obtaining p‐values and confidence intervals.
3. Effect Size vs. p‐Value
p‐Values tell you whether there’s statistically significant evidence for an effect.
Effect sizes (Cohen’s d, η 2 , R2 ) tell you how large that effect is, which can be more
meaningful in practical contexts.

By understanding each term in these formulas—what it represents and why it appears—you can
more confidently apply and interpret the corr
ChatGPT can make mistakes. Check important info.

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