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Statistics Formulas

Terms in this set (42)

Population mean
μ = ( Σ Xi ) / N
Population standard deviation =
σ = sqrt [ Σ ( Xi - μ )² / N ]
Population variance
σ² = Σ ( Xi - μ )² / N
Standardized score
Z = (X - μ) / σ
Population correlation coefficient
ρ = [ 1 / N ] * Σ { [ (Xi - μX) / σx ] * [ (Yi - μY) / σy ] }
Sample mean
x = ( Σ xi ) / n
Sample standard deviation
s = sqrt [ Σ ( xi - x )² / ( n - 1 ) ]
Sample variance
s² = Σ ( xi - x )² / ( n - 1 )
Pooled sample proportion
p = (p1 * n1 + p2 * n2) / (n1 + n2)
Variance of sample proportion =
sp² = pq / (n - 1)
Pooled sample standard deviation
sp = sqrt [ (n₁ - 1) * s₁²+ (n₂ - 1) * s₂² ] / (n₁ + n₂ - 2) ]
Pearson Correlation Coefficient
Linear correlation (sample data)
r = [ 1 / (n - 1) ] * Σ { [ (xi - x) / sx ] * [ (yi - y) / sy ] }
Linear correlation (population data) =
ρ = [ 1 / N ] * Σ { [ (Xi - μX) / σx ] * [ (Yi - μY) / σy ] }
Regression coefficient
Regression slope intercept
Combinations of n things, taken r at a time:
nCr = n! / r!(n - r)!
Probability Rule of addition
P(A or B) = P(A) + P(B) - P(A and B)
Probability Rule of multiplication
P(A and B) = P(A) P(B|A)
Probability Rule of subtraction
P(A') = 1 - P(A)
Expected value of X
E(X) = μx = Σ [ xi * P(xi) ]
Chi-square statistic
Χ² = [ ( n - 1 ) * s² ] / σ²
f statistic
f = [ s₁²/σ₁² ] / [ s₂²/σ₂² ]
Standardized test statistic
(Statistic - Parameter) / (Standard deviation of statistic)
Chi-square test statistic
Χ² = Σ[ (Observed - Expected)² / Expected ]
One-sample t-test degrees of freedom
DF = n - 1
Permutations of n things, taken r at a time
nPr = n! / (n - r)!
Mean
Percentile
z=x-µ/σ

where x is the original value,
µ =is the population mean of all values

σ is the population standard deviation of all values.

Use the Z-table to find the corresponding percentile for the standard score.
MARGIN OF ERROR FOR THE SAMPLE MEAN
...
kth percentile
1) Order all the values in the data set from smallest to largest.

2 ) Multiply k percent by the total number of values, n.

This number is called the index.

3) If the index obtained in Step 2 is not a whole number, round it up to the nearest whole number and go to Step 4a. If the index obtained in Step 2 is a whole number, go to Step 4b.

4a.Count the values in your data set from left to right (from the smallest to the largest value) until you reach the number indicated by Step 3.

The corresponding value in your data set is the kth percentile.

4b.Count the values in your data set from left to right until you reach the number indicated by Step 2.

The kth percentile is the average of that corresponding value in your data set and the value that directly follows it.
A normally distributed population--variances known
A normally distributed population--variances unknown
A population that is not normally distributed (assuming n greater than or equal to 30; the central limit theorem applies)
Hypothesis Testing Steps
(1) Data
(2) Assumptions
(3) Hypotheses
(4) Test statistic

z-score formula

(a) Distribution of test statistic
If the assumptions are correct and H0 is true, the test statistic follows the standard normal distribution.
(b) Decision rule

(5) Calculation of test statistic (Z formula)

(6) Statistical decision

(7) Conclusion
Margin of Error
Step 1: Find the critical value. The critical value is either a t-score or a z-score. If you aren't sure, see: T-score vs z-score. In general, for small sample sizes (under 30) or when you don't know the population standard deviation, use a t-score. Otherwise, use a z-score.

Step 2: Find the Standard Deviation or the Standard Error.

Multiply the critical value from Step 1 by the standard deviation or standard error from Step 2. For example, if your CV is 1.95 and your SE is 0.019, then:
1.95 * 0.019 = 0.03705
Standard Error
Margin of Error for a Proportion
phat = sample proportion ("P-hat").
n = sample size
z = z-score

Step 1: Find the p hat
Step 2: Find the z score that goes with the given confidence interval
Step 3: Insert values in formula
Step 4: Turn step 3 into a percentage
Find the P Hat
p-hat = X/n

You find p-hat by dividing the number of occurrences of the desired event by the sample size.
Cochran's Sample Size Formula
Where:

e is the desired level of precision (i.e. the margin of error),
p is the (estimated) proportion of the population which has the attribute in question,
q is 1 - p.
The z-value is found in a Z table.
Confidence Interval for a sample
Step 1: Subtract 1 from your sample size. This gives you degrees of freedom, which you'll need in step 3.

Step 2: Subtract the confidence level from 1, then divide by two.

Step 3: Look up your answers to step 1 and 2 in the t-distribution table.

Step 4: Divide your sample standard deviation by the square root of your sample size.

Step 5: Multiply step 3 by step 4.

Step 6: For the lower end of the range, subtract step 5 from the sample mean.


Step 7: For the upper end of the range, add step 5 to the sample mean.
Confidence Interval for the Mean
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