The best way to memorize formulas is by practice.
Work as hard as the bees.
SUMMARY OF FORMULAS/TESTS (CHAPTER 2 – CHAPTER 7)
 Chapter 2: Summarizing and Graphing Data
1. Construct a frequency distribution: 𝑐𝑙𝑎𝑠𝑠 𝑤𝑖𝑑𝑡ℎ ≈
(𝑚𝑎𝑥𝑖𝑚𝑢𝑚 𝑣𝑎𝑙𝑢𝑒)−(𝑚𝑖𝑛𝑖𝑚𝑢𝑚 𝑣𝑎𝑙𝑢𝑒)
𝑛𝑢𝑚𝑏𝑒𝑟 𝑜𝑓 𝑐𝑙𝑎𝑠𝑠𝑒𝑠
𝑐𝑙𝑎𝑠𝑠 𝑓𝑟𝑒𝑞𝑢𝑒𝑛𝑐𝑦
2. Relative frequency = 𝑠𝑢𝑚 𝑜𝑓 𝑎𝑙𝑙 𝑓𝑟𝑒𝑞𝑢𝑒𝑛𝑐𝑖𝑒𝑠
 Chapter 3: Statistics for Describing, Exploring, and Comparing Data
1. Sample mean: 𝑥 =
∑𝑥
𝑛
∑: the sum of a set of values
X: the individual data values
n: the number of values in a sample
2. Population mean: µ =
∑𝑥
𝑁
N: the number of values in a population
3. Median: First arrange the values from the smallest to the greatest. And then, if the
number of the values is odd, the middle value of the list is the median; if the
number of the values is even, the average value of the middle numbers is the
median.
4. 𝑚𝑖𝑑𝑟𝑎𝑛𝑔𝑒 =
𝑚𝑎𝑥𝑖𝑚𝑢𝑚 𝑣𝑎𝑙𝑢𝑒+𝑚𝑖𝑛𝑖𝑚𝑢𝑚 𝑣𝑎𝑙𝑢𝑒
2
5. range = (maximum value) − (minimum value)
6. Mean from frequency distribution: 𝑥 =
7. Weighted mean: 𝑥 =
∑(𝑓∙𝑥)
∑𝑓
∑(𝑤·𝑥)
∑𝑤
8. Sample standard deviation: 𝑠 = √
∑(𝑥−𝑥)²
𝑛−1
𝑛 ∑(𝑥 2 )−(∑ 𝑥)²
or 𝑠 = √
𝑛(𝑛−1)
∑(𝑥−µ)²
9. Population standard deviation: 𝜎 = √
10. Sample variance: 𝑠² =
∑(𝑥−𝑥)²
𝑛−1
11. Population variance: 𝜎² =
or 𝑠² =
𝑁
𝑛 ∑(𝑥 2 )−(∑ 𝑥)²
𝑛(𝑛−1)
∑(𝑥−µ)²
𝑁
12. Range rule of thumb:
For estimating a value of the standard deviation s: 𝑠 ≈
𝑟𝑎𝑛𝑔𝑒
4
For interpreting a known value of the standard deviation:
Minimum “usual” value = (mean) - 2× (standard deviation)
Maximum “usual” value = (mean) + 2× (standard deviation)
13. Coefficient of variation (or CV):
𝑠
Sample: 𝐶𝑉 = 𝑥 · 100%
Population: 𝐶𝑉 =
𝜎
µ
· 100%
14. Measures of relative standing:
Sample: 𝑧 =
𝑥−𝑥
Population: z =
𝑠
x−µ
σ
Ordinary values: -2 ≤ z score ≤ 2
Unusual values: z score < -2
or
z score > 2
15. Interquartile range (or IQR) = Q3 – Q1
Semi-interquartile range =
Midquartile=
𝑄3−𝑄1
2
𝑄3+𝑄1
2
10 – 90 percentile range = P90 – P10
 Chapter 4: probability
1. Relative frequency approximation of probability:
𝑃(𝐴) =
𝑛𝑢𝑚𝑏𝑒𝑟 𝑜𝑓 𝑡𝑖𝑚𝑒𝑠 𝐴 𝑜𝑐𝑐𝑢𝑟𝑟𝑒𝑑
𝑛𝑢𝑚𝑏𝑒𝑟 𝑜𝑓 𝑡𝑖𝑚𝑒𝑠 𝑡ℎ𝑒 𝑡𝑟𝑖𝑎𝑙 𝑤𝑎𝑠 𝑟𝑒𝑝𝑒𝑎𝑡𝑒𝑑
P: probability
A, B and C: specific events
P(A): the probability of event A occurring
2. Classical approach to probability (requires equally likely outcomes):
𝑛𝑢𝑚𝑏𝑒𝑟 𝑜𝑓 𝑤𝑎𝑦𝑠 𝐴 𝑐𝑎𝑛 𝑜𝑐𝑐𝑢𝑟
𝑃(𝐴) =
𝑛𝑢𝑚𝑏𝑒𝑟 𝑜𝑓 𝑑𝑖𝑓𝑓𝑒𝑟𝑒𝑛𝑡 𝑠𝑖𝑚𝑝𝑙𝑒 𝑒𝑣𝑒𝑛𝑡𝑠
3. Formal Addition Rule: P(A or B) = P(A) + P(B) – P(A and B)
P(A and B) is the that A and B both occur at the same time as an outcome in a trial of
procedure.
4. Rule of complementary events: P(A) + P(𝐴) = 1
5. Formal Multiplication Rule: P(A and B) = P(A) · P(B|A)
P(B|A) is the probability of event B occurring after it is assumed that event A has
already occurred. If A and B are independent events, P(B|A) is really the same as P(B).
6. Conditional probability: P(B|A)=
𝑃(𝐴 𝑎𝑛𝑑 𝐵)
𝑃(𝐴)
𝑛!
7. Permutations Rule (when Items are all different): 𝑛𝑃𝑟 = (𝑛−𝑟)!
 There are n different items available. We select r of the n items (without
replacement). We consider rearrangements of the same items to be different
sequences.
𝑛!
8. Permutations Rule(when some items are identical): 𝑛1!𝑛2!⋯𝑛𝑘!
 There are n items available and some items are identical to others. We select
all of the n items (without replacement). We consider rearrangements of
distinct items to be different sequences.
𝑛!
9. Combinations Rule: 𝑛𝐶𝑟 = (𝑛−𝑟)!𝑟!
 There are n different items available. We select r of the n items (without
replacement). We consider rearrangement of the same items to be the
same.
 Chapter 5: Discrete Probability Distributions
1. Mean for a probability distribution: µ = ∑[𝑥 · 𝑃(𝑥)]
2. Variance for a probability distribution: 𝜎² = ∑[(𝑥 − µ)² · 𝑃(𝑥)]
or 𝜎² = ∑[𝑥² · 𝑃(𝑥)] − µ²
3. Standard deviation for a probability distribution: σ = √∑[𝑥² · 𝑃(𝑥)] − µ²
4. Expected value: 𝐸 = ∑[𝑥 · 𝑃(𝑥)]
5. Binomial distribution:
Mean: µ = np
Variance: σ² = npq
Standard deviation: σ = √𝑛𝑝𝑞
𝑛!
Binomial probability: 𝑃(𝑥) = (𝑛−𝑥)!𝑥! · 𝑝 𝑥 · 𝑞 𝑛−𝑥
for x = 0, 1, 2, …, n
n: number of trials
x: number of successes among n trials
p: probability of success in any one trial
q: probability of failure in any one trial (q = 1 – p)
P(x): the probability of getting exactly x successes among the n trials
 Each trial must have all outcomes classified into two categories (commonly
referred to as success and failure).
6. Poisson distribution:
Poisson probability: P(x) =
µ𝑥 ·𝑒 −µ
𝑥!
Where e ≈ 2.71828
Mean: µ
Standard deviation: 𝜎 = √µ
 The random variable x is the number of occurrences of an event over some
interval.
 0 ≤ P(x) ≤1
 Chapter 6: Normal Probability Distributions
1. Z score formula: 𝑧 =
𝑥−µ
𝜎
(round z scores to 2 decimal places)
 When working with an individual value from a normally distributed
population, use the formula 𝑧 =
𝒙−µ
𝜎
 Other forms of the z score formula:
𝑥 = µ + (𝑧 · 𝜎)
µ=x–z·σ
.
𝜎=
𝑥−µ
𝑧
𝑧=
 X, µ and σ are known⇔
𝑥−µ
𝜎
𝑧 𝑇𝑎𝑏𝑙𝑒
z ⇔
probability
2. The Central Limit Theorem:
𝑥−µ
z score formula: 𝑧 =
𝜎
√𝑛
Standard deviation of the sample mean: σ𝑥 =
𝜎
√𝑛
 When working with a mean for some sample (or group), use the formula
=
𝒙−µ
𝜎
√𝑛
.
𝑥−µ
𝑧= 𝜎
 𝑥, µ, σ and n are known ⇔
√𝑛
𝑧 𝑇𝑎𝑏𝑙𝑒
z ⇔
probability
 Chapter 7: Estimates and Sample Sizes
Proportion
1. Estimate a population proportion:
Margin of error for proportion: 𝐸 = 𝑧α/2√
𝑝̂ 𝑞̂
𝑛
Confidence interval for population proportion p: 𝑝̂ – E < p < 𝑝̂ + E
P: population proportion
𝑥
𝑝̂ : sample proportion, 𝑝̂ =𝑛
𝑞̂: sample proportion of failures in a sample of size n, 𝑞̂ = 1 - 𝑝̂
Zα/2: critical z score based on the desired confidence level
 The sample estimate is a single value (or point) used to approximate a
population parameter.
2. Sample size for estimating Proportion p:
When 𝑝̂ is known, 𝑛 =
[𝑍α/2]² 𝑝̂𝑞̂
𝐸²
[𝑍𝛼/2]²·0.25
When 𝑝̂ is unknown, 𝑛 =
𝐸²
 If the computed sample size is not a whole number, round it up to the next
higher whole number.
3. Finding the point estimate and E from a confidence interval:
𝑃𝑜𝑖𝑛𝑡 𝑒𝑠𝑡𝑖𝑚𝑎𝑡𝑒 𝑜𝑓 𝑝: 𝑝̂ =
Margin of error: 𝐸 =
(𝑢𝑝𝑝𝑒𝑟 𝑐𝑜𝑛𝑓𝑖𝑑𝑒𝑛𝑐𝑒 𝑙𝑖𝑚𝑖𝑡)+(𝑙𝑜𝑤𝑒𝑟 𝑐𝑜𝑛𝑓𝑖𝑑𝑒𝑛𝑐𝑒 𝑙𝑖𝑚𝑖𝑡)
2
(𝑢𝑝𝑝𝑒𝑟 𝑐𝑜𝑛𝑓𝑖𝑑𝑒𝑛𝑐𝑒 𝑙𝑖𝑚𝑖𝑡)−(𝑙𝑜𝑤𝑒𝑟 𝑐𝑜𝑛𝑓𝑖𝑑𝑒𝑛𝑐𝑒 𝑙𝑖𝑚𝑖𝑡)
2
Population Mean
1. Estimate a population mean: σ known
Margin of error for mean: E = zα/2 ·
𝜎
√𝑛
Confidence interval estimate of the population mean µ: 𝑥 − 𝐸 < 𝜇 < 𝑥 + 𝐸
Sample size: 𝑛 = [
𝑍𝑎/2𝜎
𝐸
]²
2. Estimate a population mean: σ unknown but s known
Margin of error for mean: E = tα/2 ·
𝑠
√𝑛
Confidence interval estimate of the population mean µ: 𝑥 − 𝐸 < 𝜇 < 𝑥 + 𝐸
Sample size: 𝑛 = [
𝑡𝑎/2𝑠
𝐸
]²
3. Finding the point estimate and E from a confidence interval:
Point estimate of µ: 𝑥 =
Margin of error: 𝐸 =
(𝑢𝑝𝑝𝑒𝑟 𝑐𝑜𝑛𝑓𝑖𝑑𝑒𝑛𝑐𝑒 𝑙𝑖𝑚𝑖𝑡)+(𝑙𝑜𝑤𝑒𝑟 𝑐𝑜𝑛𝑓𝑖𝑑𝑒𝑛𝑐𝑒 𝑙𝑖𝑚𝑖𝑡)
2
(𝑢𝑝𝑝𝑒𝑟 𝑐𝑜𝑛𝑓𝑖𝑑𝑒𝑛𝑐𝑒 𝑙𝑖𝑚𝑖𝑡)−(𝑙𝑜𝑤𝑒𝑟 𝑐𝑜𝑛𝑓𝑖𝑑𝑒𝑛𝑐𝑒 𝑙𝑖𝑚𝑖𝑡)
2
4. Estimate a population variance:
Chi-square distribution: 𝑥² =
(𝑛−1)𝑠²
𝜎²
, refer to Table A-4, df = n – 1
Confidence interval for the population variance σ²:
(𝑛−1)𝑠²
𝑋²𝑅
< σ² <
(𝑛−1)𝑠²
𝑋²𝐿