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Statistics & Probability Formula Sheet

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Formula Sheet • Averages, Spread & Probability Laws

Statistics & Probability Formula Sheet

Essential formulas for central tendency, spread and variability (range, IQR, Tukey outlier fences, standard deviation), and probability laws (AND, OR, conditional, and expected frequency).

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Measures of Average (Central Tendency)

Arithmetic Mean (Raw Data)
xˉ=∑xn\bar{x} = \frac{\sum x}{n}

∑x = sum of all data values • n = sample size / count.

Worked Example
xˉ=3+7+5+9+65=305=6\displaystyle \bar{x} = \frac{3 + 7 + 5 + 9 + 6}{5} = \frac{30}{5} = 6
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Mean from Frequency Table
xˉ=∑(f×x)∑f\bar{x} = \frac{\sum (f \times x)}{\sum f}

f = group frequency • x = midpoint or group value • ∑f = total frequency.

Worked Example
xˉ=(2×10)+(5×20)2+5=1207≈17.14\displaystyle \bar{x} = \frac{(2 \times 10) + (5 \times 20)}{2 + 5} = \frac{120}{7} \approx 17.14
Mean, Median & Mode Calculator →
Median & Mode
Median=Midpoint,Mode=Top Frequency\text{Median} = \text{Midpoint}, \quad \text{Mode} = \text{Top Frequency}

Sort data first. Median is value at position (n + 1)/2; Mode is the most frequent.

Worked Example
1,3,5,5,9  ⟹  Median=5,  Mode=5\displaystyle 1, 3, 5, 5, 9 \implies \text{Median} = 5, \; \text{Mode} = 5
Mean, Median & Mode Calculator →

Measures of Spread & Dispersion

Data Range
Range=xmax⁡−xmin⁡\text{Range} = x_{\max} - x_{\min}

Simple measure of overall spread; sensitive to extreme anomalies.

Worked Example
Data: 2,5,7,18  ⟹  Range=18−2=16\displaystyle \text{Data: } 2, 5, 7, 18 \implies \text{Range} = 18 - 2 = 16
Range Calculator →
Interquartile Range (IQR)
IQR=Q3−Q1\text{IQR} = Q_3 - Q_1

Measures the spread of the middle 50% of sorted data; resilient to outliers.

Worked Example
Q1=4,  Q3=11  ⟹  IQR=11−4=7\displaystyle Q_1 = 4, \; Q_3 = 11 \implies \text{IQR} = 11 - 4 = 7
Range & IQR Calculator →
Quartiles Positions
Q1=n+14th,Q3=3(n+1)4thQ_1 = \frac{n+1}{4}\text{th}, \quad Q_3 = \frac{3(n+1)}{4}\text{th}

Q1 = lower quartile (25th percentile) • Q3 = upper quartile (75th percentile).

Worked Example
n=7  ⟹  Q1=2nd position,  Q3=6th position\displaystyle n=7 \implies Q_1 = 2\text{nd position}, \; Q_3 = 6\text{th position}
Range & IQR Calculator →
Outlier Check (Tukey 1.5×IQR Rule)
Low<Q1−1.5(IQR),High>Q3+1.5(IQR)\text{Low} < Q_1 - 1.5(\text{IQR}), \quad \text{High} > Q_3 + 1.5(\text{IQR})

Any observation falling strictly beyond these outer fences is flagged as an outlier.

Worked Example
Q1=10,Q3=20  ⟹  Low<10−1.5(10)=−5,  High>35\displaystyle Q_1=10, Q_3=20 \implies \text{Low} < 10 - 1.5(10) = -5, \; \text{High} > 35
Range & IQR Calculator →
Sample Standard Deviation
s=∑(x−xˉ)2n−1s = \sqrt{\frac{\sum (x - \bar{x})^2}{n - 1}}

Bessel-corrected sample standard deviation; measures spread from the mean.

Worked Example
∑(x−xˉ)2=36,  n=5  ⟹  s=364=3\displaystyle \sum (x-\bar{x})^2 = 36, \; n = 5 \implies s = \sqrt{\frac{36}{4}} = 3
Range & Spread Calculator →
Semi-Interquartile Range
SIQR=Q3−Q12=IQR2\text{SIQR} = \frac{Q_3 - Q_1}{2} = \frac{\text{IQR}}{2}

Measures the average distance of the quartiles from the median.

Worked Example
IQR=14  ⟹  SIQR=142=7\displaystyle \text{IQR} = 14 \implies \text{SIQR} = \frac{14}{2} = 7
Range & Spread Calculator →

Single Event Probability

Theoretical Probability
P(A)=n(A)n(S)=Favourable OutcomesTotal OutcomesP(A) = \frac{n(A)}{n(S)} = \frac{\text{Favourable Outcomes}}{\text{Total Outcomes}}

Valid when all sample space outcomes are equally likely; always 0 ≤ P(A) ≤ 1.

Worked Example
P(Rolling a 5)=16≈0.167\displaystyle P(\text{Rolling a 5}) = \frac{1}{6} \approx 0.167
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Experimental (Relative Frequency)
Relative Frequency=Number of Times OccurredTotal Trials\text{Relative Frequency} = \frac{\text{Number of Times Occurred}}{\text{Total Trials}}

Estimates true probability; approaches theoretical probability as trials increase.

Worked Example
14 heads in 50 flips=1450=0.28\displaystyle 14 \text{ heads in } 50 \text{ flips} = \frac{14}{50} = 0.28
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Complement Rule (Not A)
P(A′)=1−P(A)P(A') = 1 - P(A)

The sum of an event occurring and not occurring is guaranteed to equal 1.

Worked Example
P(Rain)=0.35  ⟹  P(No Rain)=1−0.35=0.65\displaystyle P(\text{Rain}) = 0.35 \implies P(\text{No Rain}) = 1 - 0.35 = 0.65
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Compound & Conditional Probability

OR Rule (Mutually Exclusive)
P(A∪B)=P(A)+P(B)P(A \cup B) = P(A) + P(B)

Applies strictly when events A and B cannot both occur at the same time (mutually exclusive).

Worked Example
P(1 or 2)=16+16=26=13\displaystyle P(1 \text{ or } 2) = \frac{1}{6} + \frac{1}{6} = \frac{2}{6} = \frac{1}{3}
Probability Calculator →
General Addition Rule (Overlap)
P(A∪B)=P(A)+P(B)−P(A∩B)P(A \cup B) = P(A) + P(B) - P(A \cap B)

Subtracting the intersection P(A ∩ B) prevents double-counting overlapping events.

Worked Example
P(Ace or Heart)=452+1352−152=1652=413\displaystyle P(\text{Ace or Heart}) = \frac{4}{52} + \frac{13}{52} - \frac{1}{52} = \frac{16}{52} = \frac{4}{13}
Probability Calculator →
AND Rule (Independent Events)
P(A∩B)=P(A)×P(B)P(A \cap B) = P(A) \times P(B)

Applies when the outcome of event A does not influence the outcome of event B.

Worked Example
P(Head and Head)=12×12=14=0.25\displaystyle P(\text{Head and Head}) = \frac{1}{2} \times \frac{1}{2} = \frac{1}{4} = 0.25
Probability Calculator →
Conditional Probability
P(A∣B)=P(A∩B)P(B)(P(B)>0)P(A \mid B) = \frac{P(A \cap B)}{P(B)} \quad (P(B) > 0)

The probability of event A occurring given that event B has already occurred.

Worked Example
P(A∩B)=0.2,  P(B)=0.5  ⟹  P(A∣B)=0.20.5=0.4\displaystyle P(A \cap B) = 0.2, \; P(B) = 0.5 \implies P(A \mid B) = \frac{0.2}{0.5} = 0.4
Probability Calculator →
Dependent Events (Without Replacement)
P(A∩B)=P(A)×P(B∣A)P(A \cap B) = P(A) \times P(B \mid A)

Used for tree diagrams where removing items reduces the remaining sample pool.

Worked Example
P(2 Red from 3R/7B)=310×29=690=115\displaystyle P(\text{2 Red from 3R/7B}) = \frac{3}{10} \times \frac{2}{9} = \frac{6}{90} = \frac{1}{15}
Probability Calculator →
Expected Frequency
E(X)=n×P(A)E(X) = n \times P(A)

n = number of trials • P(A) = probability of success on a single trial.

Worked Example
E(Sixes in 300 rolls)=300×16=50\displaystyle E(\text{Sixes in 300 rolls}) = 300 \times \frac{1}{6} = 50
Probability Calculator →
Curriculum & Revision Note

About These Statistics & Probability Formulas

Statistics and data handling questions are heavily tested across secondary school maths and sciences. This sheet covers averages (arithmetic mean, grouped frequency mean, median, mode), spread and dispersion (range, IQR, Tukey 1.5×IQR outlier boundaries, standard deviation), alongside probability addition/multiplication laws and expected frequency.

Need step-by-step solutions? Open Mean, Median & Mode Calculator →