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Statistics Equations & Formulas

Calculations

Chapter 2

Frequency Distributions

  • Consider the following data representing the number of miles professors drive to work each day: 3.8, 2.7, 9.3, 6.5, 5.8, 7, 10.2, 1, 3.7, 9.1, 6.2, 11, 11.9, 5.5, 4.8, 7.3, 9.1, 1.4
Number of Miles Professors Drive to Work Each Day

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Number of Miles Professors Drive to Work Each Day
Class Frequency Class Boundaries Midpoint Relative Frequency Cumulative Frequency
Lower Upper
Calculation N/A Count the numbers in the range
upper limit (lower class) + lower limit (upper class) 2
upper limit + lower limit 2
frequency sample size
Add frequencies of previous classes
1.0-2.9 2.7, 1, 1.4 (3)
0.9+1 2
= 0.95
3.0+2.9 2
= 2.95
1 + 2.9 2
= 1.95 		3 18 = 0.17 = 17% 3
3.0-4.9 3.8, 3.7, 4.8 (3)
3.0+2.9 2
= 2.95
5.0+4.9 2
= 4.95
3+4.9 2
= 3.95 		3 18 = 0.17 = 17% 3 + 3 = 6
5.0-6.9 6.5, 5.8, 6.2, 5.5 (4)
5.0+4.9 2
= 4.95
7.0+6.9 2
= 6.95
5+6.9 2
= 5.95 		4 18 = 0.22 = 22% 6 + 4 = 10
7.0-8.9 7, 7.3 (2)
7.0+6.9 2
= 6.95
9.0+8.9 2
= 8.95
7.0+8.9 2
= 7.95 		2 18 = 0.11 = 11% 10 + 2 = 12
9.0-10.9 9.3, 10.2, 9.1, 9.1 (4)
9.0+8.9 2
= 8.95
11+10.9 2
= 10.95
9+10.9 2
= 9.95 		4 18 = 0.22 = 22% 12 + 4 = 16
11.0-12.9 11, 11.9 (2)
11+10.9 2
= 10.95
13+12.9 2
= 12.95
11+12.9 2
= 11.95 		2 18 = 0.11 = 11% 16 + 2 = 18

Chapter 3

Measures of Center

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Measures of Center
Measure Good For Sample Population Calculation
Mean Symmetrical ̄ μ • Stat > Edit
• 1: Edit…
• Input Data
• Stat > Calc
• 1: 1-Var Stats
• List: Input List
• ̄
Weighted Mean Symmetrical w/ Weights ̄ μ • Stat > Edit
• 1: Edit…
• Input Data
• Stat > Calc
• 1: 1-Var Stats
• List: Input List
• FreqList: Weight List
• ̄
Median Skewed & Outliers Med Med • See “Mean”
• Value: Med
Mode Qualitative N/A N/A • Stat > Edit
• 1: Edit…
• Input Data
• Stat > Edit
• 2: SortA(
• 2nd > List #
• Mode = Most Often
Measures of Dispersion

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Measures of Dispersion
Measure Good For Sample Population Formula / Calculation
Standard Deviation Symmetrical s σ • See Mean
• Value: Sx or σx

√ variance
Variance Symmetrical ² σ² • See Mean
• Vars > Vars
• 5: Statistics
• 3: Sx or 4: σx
• SD²
Range Easy Calculations N/A N/A max − min
IQR Skewed & Outliers N/A N/A • See Mean
• Values: Q₁ and Q₃
• Q₃ − Q₁

Box Plots

  • Five Number Summary (1-Var Stats)
    •  Minimum, Q1, Median, Q3, Maximum
    • Potential Outliers
      • Lower Limit: 𝑄1−(1.5∗𝐼𝑄𝑅)
      • Upper Limit: 𝑄3+(1.5∗𝐼𝑄𝑅)
Z-Scores
Sample Population
z = x − ̄ s z = x − μ σ
Percentiles
Location Percentile in a List Empirical Rule
Equation l = n × P 100 P = l n × 100 • 68% in 1 SD
• 95% in 2 SD
• 99.7% in 3 SD
Decimal Rules If the answer is a decimal, round up.

If the answer is a whole number, find the mean of the number in that location and the number in the next location up. 		N/A N/A
Example:
6.8, 9.1, 8.7, 7.5, 8.2,
5.4, 6.5, 8.5, 7.3, 6.6,
5.9, 7.3, 9.3, 7.4 		• 12th percentile
• Order numbers from least to greatest.
• l = 14 × 12 100 = 1.68 ≅ 2
• Find the second number in the list: 5.9 		• What percentile is 8.2?
• Order the numbers from least to greatest.
• Find where 8.2 is in the list.
• P = 10 14 × 100 = 71st 		N/A
Graph N/A N/A The image depicts a standard normal distribution curve, which is shaped like a symmetric bell. The curve is colored in a light blue shade, with the peak situated at the center, labeled as "Mean." Below the curve, the x-axis is marked with a range of values from -3 to 3, with the central value at 0. Vertical lines extend down from the curve at points -3, -2, -1, 1, 2, and 3. Each of these points is accompanied by percentages indicating the proportion of values that fall within this range under the curve. The percentages are as follows: 0.15% at -3 and 3, 2.35% at -2 and 2, and 13.5% at -1 and 1, with 34% filling the area between -1 and 1.

Chapter 4

Experimental/Empirical Probability


P(E) = f n = frequency sample size


Classical/Theoretical Probability


  • P(E) = n(E) n(S) = number of outcomes in the event number of total outcomes
  • 0 ≤ P(E) ≤ 1

  • Complement (NOT)
    • P(Ec) = 1 − P(E)      P(Ec) + P(E) = 1      P(E) = 1 − P(Ec)

  • Addition (OR)
    • Mutually Exclusive Events
      (a) P(E or F) = P(E) + P(F)     OR     P(E ∪ F) = P(E) + P(F)
    • Not Mutually Exclusive Events
      (a) P(E or F) = P(E) + P(F) − P(E and F)     OR     P(E ∪ F) = P(E) + P(F) − P(E ∩ F)

  • Multiplication (AND)
    • Independent Events
      (a) P(E and F) = P(E) × P(F)     OR     P(E ∩ F) = P(E) × P(F)
    • Not Independent Events
      (a) P(E and F) = P(E) × P(F|E) = P(F) × P(E|F)     OR     P(E ∩ F) = P(E) × P(F|E) = P(F) × P(E|F)

  • Conditional (Given That/If)
    • P(F|E) = P(E and F) P(E)     OR     P(F|E) = (∩F) P(E)

Fundamental Counting Principle


  • k1 × k2 × k3 … kn     multiply the outcomes of a multistage event to find the total possible outcomes.

Chapter 5

Discrete Probability Distributions

  • Key Phrases and Graphs
    • Expected value
    • Expected

    x P(x)
  • Calculator: See Weighted Means
    • P(x) will be the FreqList
    • P(x) will add to 1
  • Expected Value: ̄
  • Standard Deviation: σ
  • Variance: σ2

Binomial Probability

Conditions Probability Calculator Input Key Phrases Equations
• Simple Random Sample
• Independent Trials
• Probability of Success = p
• Number of Trials = n 		P(X = x)
• 2nd > Vars
• A: binompdf
• 2nd > Vars
• B: binomcdf
binpdf n,p,x • Exactly x μ = np

σ2 = np(1 − p)

σ = √np(1 − p)
P(X ≤ x) bincdf n,p,x • At most x
• x or fewer
• x or less
P(X < x) bincdf n,p,(x-1) • Less than x
• Fewer than x
P(X ≥ x) 1-bincdf n,p,(x-1) • At least x
• x or more
P(X > x) 1-bincdf n,p,x • Greater than x
• More than x

Chapter 6

Normal Probability

  • Find
    • Area under the curve, probability, percentile, percent
  • Standard Normal Distribution
    • Mean = 0     SD = 1
Conditions Probability Calculator Input Key Phrases
• Continuous Data
• Symmetrical
• Mean in Middle
• Data Separated by Standard Deviation
• Area Adds to 1 		• P(X < x)
• P(X ≤ x) 		• 2nd > Vars
• 2: normalcdf 		• Lower: -1E99
• Upper: X
• Mean: μ
• SD: σ 		• Less Than
• At Most
• Fewer Than
• X or Less
• P(X > x)
• P(X ≥ x) 		• Lower: X
• Upper: 1E99
• Mean: μ
• SD: σ 		• Greater Than
• At Least
• More Than
• X or More
• P(X ≤ x ≤ Y)
• P(X < x < Y) 		• Lower: X
• Upper: Y
• Mean: μ
• SD: σ 		• Between
• P(x < X or x > Y)
• P(x ≤ X or x ≥ Y) 		1 − “Between” • Tails
• Less than X or Greater than Y
Differs by Less • Lower: μ − X
• Upper: μ + X
• Mean: μ
• SD: σ 		• Differs by Less Than X
• Differs by Fewer Than X
Differs by More • 1 – “Differs by Less”
• 2 * normalcdf a tail 		• Differs by Greater Than X
• Differs by More Than X

invNorm

  • Find
    • z-score
    • ѱ𲹲ܰ𳾱Գٲ
Conditions Direction Calculator Input Key Phrases
TI-83 TI-84
• Continuous Data
• Symmetrical
• Mean in Middle
• Data Separated by Standard Deviation
• Area Adds to 1 		Left • 2nd > Vars
• 3: invNorm 		p, μ, σ • Area: p
• Mean: μ
• SD: σ
• Left 		• To the Left
• Bottom %
• Percentile
• Less Than
Right (1-p), μ, σ • Area: p
• Mean: μ
• SD: σ
• Right 		• To the Right
• Top %
• More Than
Between -z and z • 1-p 2 , 0,1 Gives -z
• z = -z × -1 		• Area: p
• Mean: 0
• SD: 1
• Center 		• Between
• This is ONLY z-scores
Left -z and Right z • p 2 , 0,1 Gives -z
• z = -z × -1 		• Area: 1-p
• Mean: 0
• SD: 1
• Center 		• Left of -z Plus Right of z
• This is ONLY z-scores

Chapter 7

Central Limit Theorem: Probability 

  • ALL CLT probability questions follow the normal distribution.
  • What are you being asked about?
Individual Mean Proportion
Mean μ μ p
Standard Deviation σ σ √n √ p(1-p) n
Conditions • Normal • n ≥ 30
• Normal 		• np ≥ 10
• n(1 − p) ≥ 10
• Normal
z-score z = ̄ − μ σ z = ̄ − μ σ √n z = ̂ − p √ p(1-p) n
Other Equations N/A N/A • ̂ = x n

• x = ̂ × n
Key Phrases: Probability • Individual
• Random ____ Chosen 		• Mean • Proportion
• Percent
• x of n
  • Does it say, “differs by”?
No Differs By Less Differs By More
Individual Mean Proportion
Calculator
Functions 		Follow
normalcdf
Rules 		• μ − x
• μ + x
• Mean: μ
• SD: σ 		• μ − x
• μ + x
• Mean: μ
• σ √n 		• p − %
• p + %
• Mean: p
• √ p(1-p) n 		• 1 − “Differs by Less”
• 2 × normalcdf (one of the sides)
Diagram N/A A gray shaded normal distribution curve centered at zero, with horizontal axis markings from -3 to 3. A bell-shaped curve representing a standard normal distribution, with shaded areas in the tails extending from -3 to -2 and 2 to 3.

Chapters 8 and 9

Confidence Intervals
Mean Proportion Two Means
σ Known σ Unknown
Formula (̄ − E, ̄ + E) (̄ − E, ̄ + E) (̂ − E, ̂ + E) (̄₁ − ̄₂) − E,
(̄₁ − ̄₂) + E)
Calculator Function Z Interval T Interval 1-PropZInt 2-SampZInt
Margin of Error E = z α 2 σ √n E = t α 2 s √n E = z α 2 √ ̂(1 − ̂) n E = z α 2 √ σ₁² ₁ + σ₂² ₂
Sample Size n = ( z α 2 × σ E ) 2 N/A n = p(1 − p) ( z α 2 E ) 2 N/A
Critical Value zα/2 tα/2 zα/2 zα/2
Requirements
  • n ≥ 30
  • Approximately Normal
  • n ≥ 30
  • Approximately Normal
  • np ≥ 10
  • n(1 − p) ≥ 10
  • Approximately Normal
  • n1 ≥ 30 and
    n2 ≥ 30
  • Both Populations Approximately Normal
Key Phrases
  • Interval
  • Endpoints
  • Mean
  • Population Standard Deviation Known
  • Interval
  • Endpoints
  • Mean
  • Population Standard Deviation Unknown
  • Sample Standard Deviation
  • Interval
  • Endpoints
  • Proportion
  • Interval
  • Endpoints
  • Difference Between Two Means
  • Population Standard Deviations Known
Extra Equations • E = upper endpoint − lower endpoint 2
• μ = upper endpoint + lower endpoint 2 		• E = upper endpoint − lower endpoint 2
• μ = upper endpoint + lower endpoint 2 		• E = upper − lower 2
• p = upper + lower 2
• x = ̂ × n 		• E = upper − lower 2
• μ1 − μ2 = upper + lower 2
Critical Value Calculations
Z Interval T Interval Proportion Interval 2 Sample Z Interval
Symbol zα/2 tα/2 zα/2 zα/2
General
  • 2nd > Vars
  • invNorm
  • Stat > Tests
  • 8: T Interval
  • Stat
  • 2nd > Vars
  • invT
 Same as Z Interval Same as Z Interval
TI-83 ( 1 − c 2 ) , 0,1
  • ̄: 0
  • Sx: √n
  • n: Sample Size
  • C-Level: c
 N/A
TI-84
  • Area: 1 − c 2
  • Mean: 0
  • SD: 1
  • Area: 1 − c 2
  • DF: n − 1
Endpoint Calculations
Z Interval T Interval Proportion Interval 2 Sample Z Interval
General
  • Stat > Tests
  • 7: Z Interval
  • Stat if Numbers
  • Data if List (Stat > Edit)
  • Stat > Tests
  • 8: T Interval
  • Stat if Numbers
  • Data if List (Stat > Edit)
  • Stat > Tests
  • A: 1-PropZInt
  • Stat > Tests
  • 9: 2-SampZInt
  • Stat if Numbers
  • Data if Lists (Stat > Edit)
Calculator Input
  • σ: Population SD
  • x̄: Sample Mean
  • n: Sample Size
  • C-Level: c
  • x̄: Sample Mean
  • Sx: Sample SD
  • n: Sample Size
  • C-Level: c
  • x: Number of Successes
  • n: Sample Size
  • C-Level: c
  • σ1: Population SD Group 1
  • σ2: Population SD Group 2
  • x̄1: Sample Mean Group 1
  • n1: Sample Size Group 1
  • x̄2: Sample Mean Group 2
  • n2: Sample Size Group 2
  • C-Level: c

Chapters 10 & 11

“
Hypothesis Tests
Mean Proportion Two Means
σ Known σ Unknown
Calculator Function Z-Test T-Test 1-PropZTest 2-SampZTest
Alternative Hypotheses
  • u ≠ u
  • u < μ
  • μ > μ
  • u ≠ u
  • u < μ
  • μ > μ
  • p ≠ p
  • p < p
  • p > p
  • μ1 ≠ μ2
  • μ1 < μ2
  • μ1 > μ2
Test Statistic z = ̄ − μ σ √n t = ̄ − μ s √n z = ̂ − p √ p(1 − p) n z = (̄1 − ̄2) − (μ1 − μ2) √ σ12 n1 + σ22 n2
Requirements
  • n ≥ 30
  • Approximately Normal
  • n ≥ 30
  • Approximately Normal
  • np ≥ 10
  • n(1-p) ≥ 10
  • Approximately Normal
  • n1 ≥ 30 and n2 ≥ 30
  • Both Populations Approximately Normal
Key Phrases
  • Test
  • Claim
  • Sufficient Evidence
  • Mean
  • Population Standard Deviation Known
  • Test
  • Claim
  • Sufficient Evidence
  • Mean
  • Population Standard Deviation Unknown
  • Sample Standard Deviation
  • Test
  • Claim
  • Sufficient Evidence
  • Proportion
  • Test
  • Claim
  • Sufficient Evidence
  • Difference Between Two Means
  • Population Standard Deviation Known
Conclusions p − value ≤ α Reject the null hypothesis.
There is sufficient evidence to support the alternative hypothesis.
p − value > α Do not reject the null hypothesis.
There is not sufficient evidence to support the alternative hypothesis.
Test Calculations
Z Test T Test Proportion Test 2 Sample Z Test
General
  • Stat > Tests
  • 1: Z-Test
  • Stat if Numbers
  • Data if List (Stat > Edit)
  • Stat > Tests
  • 2: T-Test
  • Stat if Numbers
  • Data if List (Stat > Edit)
  • Stat > Tests
  • 5: 1-PropZTest
  • Stat > Tests
  • 3: 2-SampZTest
  • Stat if Numbers
  • Data if Lists (Stat > Edit)
Calculator Input
  • μ0: Null Hypothesis
  • σ: Population SD
  • ̄: Sample Mean
  • n: Sample Size
  • μ: ≠ μ0, < μ0, > μ0
    Alternative Hypothesis
  • μ0: Null Hypothesis
  • ̄: Sample Mean
  • Sx: Sample SD
  • n: Sample Size
  • μ: ≠ μ0, < μ0, > μ0
    Alternative Hypothesis
  • p0: Null Hypothesis
  • x: Number of Successes
  • n: Sample Size
  • prop: ≠ p0, < p0, > p0
    Alternative Hypothesis
  • σ1: Population SD Group 1
  • σ2: Population SD Group 2
  • ̄1: Sample Mean Group 1
  • n1: Sample Size Group 1
  • ̄2: Sample Mean Group 2
  • n2: Sample Size Group 2
  • μ1 ≠ μ2 < μ2 > μ2
    Alternative Hypothesis

Chapter 12

Linear Regression
Explanatory Variable Response Variable Slope y-intercept Correlation Coefficient Coefficient of Determination
Symbol x y a b r r2
Purpose Predict changes in y Respond to changes in x For every one x, y changes by b If x = 0, y = a
  • Close to -1 or 1: Strong Relationship
  • Close to 0: Weak Relationship
  • Sign Matches Correlation
    (+ or -)
 Tells how much of y is predicted by x
AKA • Independent
• Predictor 		• Dependent
• Predicted 		• Rate of Change • Starting Value • Strength and Direction of Relationship • Accuracy of Data
Calculator Function • Stat > Edit
• Input Data
• Stat > Calc
• 4: LinReg(ax+b) 		• Xlist: List With x Data
• Ylist: List With y Data
• FreqList: Blank
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