CBSE Class 8 Mathematics Statistics and Probability Notes Set 15

Mean

The mean (arithmetic mean) of n observations is their sum divided by n. If the values x₁, x₂, ..., xₖ occur with frequencies f₁, f₂, ..., fₖ, the mean is Σ fx / Σ f.
The sum of the deviations of the observations from their mean is always zero. If every observation is increased by a, the mean increases by a; if every observation is multiplied by a, the mean is multiplied by a.

Median and Mode

Median: arrange the data in order. For an odd number n of values it is the middle value, the ((n + 1)/2)th. For an even number n it is the average of the (n/2)th and (n/2 + 1)th values. Mode: the value that occurs most often.
For moderately skewed data the empirical relation 3 × Median = Mode + 2 × Mean can be used to find the third measure when two are known.

Probability of an Event

In a random experiment with equally likely outcomes, P(E) = (number of outcomes favourable to E) / (total number of outcomes). Always 0 ≤ P(E) ≤ 1. A sure event has probability 1, an impossible event has probability 0, and P(not E) = 1 − P(E).
The sum of the probabilities of all the elementary outcomes of an experiment is 1.

Standard Sample Spaces

One coin: 2 outcomes. Two coins: 4 outcomes. Three coins: 8 outcomes. One die: 6 outcomes. Two dice: 36 outcomes (sum 7 can occur in 6 ways, the most likely sum). A pack of cards: 52 cards, 26 red and 26 black, 4 suits of 13, with 12 face cards (4 kings, 4 queens, 4 jacks) and 4 aces.
An ordinary year has 52 weeks and 1 extra day, so the chance of 53 Sundays is 1/7. A leap year has 2 extra days, so the chance of 53 Sundays is 2/7.

Worked Examples

Example 1: Mean of the data 2, 4, 6, 8, 10 is 30/5 = 6. If each value is multiplied by 3 the mean becomes 18.
Example 2: Median of 12, 7, 9, 15, 10, 20 (even count): in order 7, 9, 10, 12, 15, 20, so the median is (10 + 12)/2 = 11.
Example 3: A die is thrown. The probability of a number greater than 4 is 2/6 = 1/3, and the probability of not getting a multiple of 3 is 1 − 2/6 = 2/3.
Example 4: Two coins are tossed. The probability of at least one head is 1 − P(no head) = 1 − 1/4 = 3/4.