Statistics

1. Arithmetic Mean

Definition:

The arithmetic mean is the average of all values in a dataset. It represents the central value.

Formulas:

• Individual Series: $\bar{x} = \dfrac{\sum x}{n}$

• Discrete Series: $\bar{x} = \dfrac{\sum fx}{\sum f}$

Where:

• $x$ = value
• $f$ = frequency
• $n$ = total number of values

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2. Median

Definition:

The median is the middle value when data is arranged in ascending or descending order.

Formulas:

• Individual Series:

  If $n$ is odd: Median = middle value

  If $n$ is even: Median = average of two middle values

• Discrete Series:

  Find cumulative frequency (CF), then locate the value where $\text{CF} \geq \dfrac{n}{2}$

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3. Mode

Definition:

The mode is the value that appears most frequently in a dataset.

Method:

• Individual Series: Identify the value with highest frequency.

• Discrete Series: Value corresponding to highest frequency.

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4. Quartiles

Definition:

Quartiles divide a dataset into four equal parts.

• $Q_1$ = First Quartile (25% of data below it)

• $Q_2$ = Second Quartile = Median

• $Q_3$ = Third Quartile (75% of data below it)

Formulas:

• Individual Series: $Q_k = \text{Value at position } \dfrac{k(n+1)}{4}$, where $k = 1, 2, 3$

• Discrete Series:
  Use cumulative frequency to locate:

  • $Q_1$: $\dfrac{n}{4}$

  • $Q_2$: $\dfrac{n}{2}$

  • $Q_3$: $\dfrac{3n}{4}$

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Relationships & Differences

Measure	Focus	Sensitive to Outliers	Best Use Case
Mean	Average value	Yes	Balanced data
Median	Middle value	No	Skewed data or outliers present
Mode	Most frequent	No	Categorical or repeating data
Quartiles	Spread of data	No	Understanding distribution

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Real-World Examples

• Mean: Average marks of students in a class.

• Median: Median income in a country to avoid skew from billionaires.

• Mode: Most common shoe size sold in a store.

• Quartiles: Used in box plots to analyze exam score distribution.

When to Use Each Measure

• Mean vs. Median:
  Suppose a company has 10 employees earning $30,000 each and 1 CEO earning $1,000,000. The mean salary will be much higher than what most employees earn, while the median salary will better represent the typical employee's pay. In this case, median is preferred over mean.

• Mode vs. Mean/Median:
  In a survey of favorite ice cream flavors, the most popular flavor (mode) is more meaningful than the mean or median, which may not correspond to any actual flavor. Use mode for categorical data.

• Quartiles vs. Mean:
  When analyzing exam scores, quartiles help identify the spread and detect outliers, while the mean only gives the average. Quartiles are useful for understanding variability and distribution.

• Mean:
  For consistent, symmetric data without outliers (like heights of adult men in a population), the mean is a reliable measure of central tendency.

• Median:
  For house prices in a city (where a few very expensive homes can skew the average), the median gives a better sense of a "typical" price.

• Mode:
  In manufacturing, the mode can indicate the most common defect type, helping to target quality improvements.

• Quartiles:
  In finance, quartiles are used to compare investment returns and assess risk by looking at the spread between the lower and upper quartiles.

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Probability

Definition & Meaning

Probability is the measure of the likelihood that a particular event will occur. It ranges from 0 (impossible event) to 1 (certain event).

• If an event is certain, its probability is 1.

• If an event is impossible, its probability is 0.

• All probabilities lie between 0 and 1.

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Key Terms

Term	Meaning
Experiment	A process that leads to an outcome (e.g., tossing a coin)
Trial	Each repetition of an experiment
Outcome	A possible result of an experiment
Sample Space (S)	The set of all possible outcomes
Event (E)	A subset of the sample space; the outcome(s) we are interested in
Favorable Outcomes	Outcomes that satisfy the condition of the event

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Formula

$\text{Probability of an event } E = P(E) = \dfrac{\text{Number of favorable outcomes}}{\text{Total number of possible outcomes}}$

Examples:

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Experiment: Tossing a coin

Sample Space: $S = \{ Head, Tail \}$

Event: Getting a Head

Favorable outcomes = 1

Total outcomes = 2

Probability: $P(E) = \dfrac{1}{2}$

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Experiment: Rolling a fair die

Sample Space: $S = \{ 1, 2, 3, 4, 5, 6 \}$

Event: Getting an even number

Favorable outcomes = 6 → 3 outcomes

Total outcomes = 6

Probability: $P(E) = \dfrac{3}{6} = \dfrac{1}{2}$

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Experiment: Drawing a card from a deck of 52 cards

Event: Getting a King

Favorable outcomes = 4 (one King in each suit)

Total outcomes = 52

Probability: $P(E) = \dfrac{4}{52} = \dfrac{1}{13}$

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Properties of Probability

$0 \leq P(E) \leq 1$

$P(\text{Impossible event}) = 0$

$P(\text{Certain event}) = 1$

$P(E') = 1 - P(E) \quad \text{(Complement of event E)}$

Real-Life Applications

• Predicting weather (chance of rain)

• Insurance risk assessment

• Quality control in manufacturing

• Games of chance (dice, cards, coins)

• Medical testing (probability of disease presence)

Probability Scale & Empirical Probability

Probability Scale

The probability scale is a number line from 0 to 1 that shows how likely an event is to occur.

Probability Value	Description	Example
$0$	Impossible Event	Getting a 7 on a standard die
$0.5$	Equally Likely	Getting Head or Tail in a fair coin toss
$1$	Certain Event	Getting a number less than 7 on a die

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Empirical Probability (Experimental Probability)

Empirical Probability is the probability based on actual experiments or observations. It is calculated using the ratio of the number of times an event occurs to the total number of trials.

Formula:

$P(E) = \dfrac{\text{Number of times event E occurs}}{\text{Total number of trials}}$

Examples:

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A coin is tossed 100 times. It lands on heads 56 times.
Find the empirical probability of getting heads.

P(Head) = $\dfrac{56}{100} = 0.56$

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A die is rolled 60 times. The number 4 appears 9 times.
Find the empirical probability of getting a 4.

P(4) = $\dfrac{9}{60} = 0.15$

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In a class of 40 students, 28 like football.
Find the probability that a randomly selected student likes football.

P(Football) = $\dfrac{28}{40} = 0.7$

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Key Differences: Theoretical vs. Empirical Probability

Aspect	Theoretical Probability	Empirical Probability
Basis	Assumes all outcomes are equally likely	Based on actual experiments or observations
Formula	$P(E) = \dfrac{\text{Favorable outcomes}}{\text{Total outcomes}}$	$P(E) = \dfrac{\text{Observed frequency of E}}{\text{Total trials}}$
Example	Probability of rolling a 6 on a fair die = $\dfrac{1}{6}$	If 6 appears 8 times in 50 rolls: $\dfrac{8}{50} = 0.16$
Accuracy	Idealized, assumes fairness and perfect conditions	Reflects real-world variation and randomness
Use Case	Used in games, theoretical models, and simulations	Used in surveys, experiments, and statistical analysis
Affected by Bias	No (assumes fairness)	Yes (depends on sample and conditions)