Statistics Exercises

Mean

Example (Individual):

Marks: 10, 20, 30, 40, 50

$\bar{x} = \dfrac{10 + 20 + 30 + 40 + 50}{5}$

= $30$

Example (Discrete):

Marks ($x$)	Frequency ($f$)
10	2
20	3
30	5

$\bar{x}$

= $\dfrac{(10 \cdot 2 + 20 \cdot 3 + 30 \cdot 5)}{2 + 3 + 5}$

= $\dfrac{10 + 60 + 150}{10}$

= $24$

Median

Example (Individual):

Data: 5, 10, 15, 20, 25

Median = 15 (middle value)

Example (Discrete):

Value ($x$)	Frequency ($f$)	Cumulative Frequency
10	2	2
20	3	5
30	5	10

Total number of values, $n = \sum f = 10$

To find the median:

• Compute $\dfrac{n}{2} = \dfrac{10}{2} = 5$

Now, look for the cumulative frequency just greater than or equal to 5.

From the table:

• Cumulative frequency for 10 is 2 (less than 5)

• Cumulative frequency for 20 is 5 (equal to 5)

• Cumulative frequency for 30 is 10 (greater than 5)

So, the median is the value corresponding to the cumulative frequency where $\text{CF} \geq 5$, which is 20.

Final Answer: $\boxed{20}$ is the median.

Mode

Example (Individual):

Data: 2, 3, 3, 5, 7

Mode = 3 (frequency = 2)

Example (Discrete):

Value ($x$)	Frequency ($f$)
10	2
20	5
30	3

Mode = 20 (highest frequency)

Quartiles

Example (Individual):

Data: 5, 10, 15, 20, 25, 30, 35

$n = 7$

$Q_1$ = value at $\dfrac{1(7+1)}{4} = 2^{nd}$ → 10

$Q_2$ = Median = 20

$Q_3$ = value at $\dfrac{3(7+1)}{4} = 6^{th}$ → 30

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Probability Exercises

Formula Reminder

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

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Example 1: Rolling a Die

Question: What is the probability of getting a number greater than 4 when rolling a fair die?

• Sample Space: $S = \{1, 2, 3, 4, 5, 6\}$
• Favorable outcomes: $\{5, 6\}$ → 2 outcomes
• Total outcomes: 6

Solution:
$P(\text{number} > 4) = \dfrac{2}{6} = \dfrac{1}{3}$

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Example 2: Drawing a Card

Question: What is the probability of drawing a red queen from a standard deck of 52 cards?

• Total red queens: 2 (Queen of Hearts, Queen of Diamonds)
• Total cards: 52

Solution:
$P(\text{red queen}) = \dfrac{2}{52} = \dfrac{1}{26}$

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Example 3: Tossing Two Coins

Question: What is the probability of getting exactly one head?

• Sample Space: $S = \{HH, HT, TH, TT\}$
• Favorable outcomes: $\{HT, TH\}$ → 2 outcomes
• Total outcomes: 4

Solution:
$P(\text{exactly one head}) = \dfrac{2}{4} = \dfrac{1}{2}$

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Example 4: Spinner Game

Question: A spinner is divided into 5 equal sections labeled A, B, C, D, E. What is the probability of landing on A or E?

• Favorable outcomes: $\{A, E\}$ → 2 outcomes
• Total outcomes: 5

Solution:
$P(A \text{ or } E) = \dfrac{2}{5}$

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Example 5: Real-World – Student Survey

Question: In a class of 40 students, 28 like football. What is the probability that a randomly selected student likes football?

• Favorable outcomes: 28
• Total outcomes: 40

Solution:
$P(\text{likes football}) = \dfrac{28}{40} = 0.7$

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Example 6: Empirical Probability – Dice Roll

Question: A die is rolled 60 times. The number 3 appears 12 times. What is the empirical probability of getting a 3?

• Observed frequency: 12
• Total trials: 60

Solution:
$P(3) = \dfrac{12}{60} = 0.2$

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Exercises

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Stats

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Exercise 1

Q1.

$x$	10	20	30
$f$	2	3	5

Show solution

$\displaystyle \bar{x} = \dfrac{\sum{fx}}{\sum{f}} \quad\text{where }f\text{ is the frequency}$

$= \dfrac{(10 \cdot 2 + 20 \cdot 3 + 30 \cdot 5)}{2 + 3 + 5}$

$= \dfrac{260}{10} = 26$

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Probability

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Exercise 1

Q1. What is the probability that the coin comes out in a game of die?

i. If 1 brings the coin out

ii. If six brings the coin out

iii. If both 1 and 6 brings the coin out?

Details

Show solution

Total possible outcomes = 6 (faces of a fair die).

i. If 1 brings the coin out: favorable case(s) = 1

$P = \dfrac{1}{6}$

ii. If 6 brings the coin out: favorable case(s) = 1

$P = \dfrac{1}{6}$

iii. If both 1 and 6 bring the coin out: favorable = 2

$P = \dfrac{2}{6} = \dfrac{1}{3}$

Exercise 2

Q2. What is the probability that the card is Ace (A) if a card is drawn randomly from a deck?

Show solution

Total possible outcomes = 52.

i. Favorable cases = 4 (an Ace of each suit)

$P = \dfrac{4}{52}$

Thus, $P = \dfrac{1}{13}$

Exercise 3

Q3. What is the probability that the card is face card if a card is drawn randomly from a deck?

Show solution

Total possible outcomes = 52.

i. Favorable cases = 3 * 4 (Jack, Queen and King of each suit)

$P = \dfrac{4 * 3}{52}$

Thus, $P = \dfrac{3}{13}$