Philippine Christian University

Sampaloc 1, Dasmariñas City, Cavite 4114

SENIOR HIGH SCHOOL
S.Y. 2024-2025

STATISTICS AND PROBABILITY (GRADE 11)

NUMERO CLUB REVIEWER

WEEK 1

• Statistics- the science of the development of applications

• Probability- is branch of mathematics that deals with uncertainty, estimation of how likely
• Variable - describes a person, place, thing, or idea, characteristic that is observable or measurable
• Data collection- process of gathering and measuring information on variables
• Sample Space- set of all possible outcome
• Event- subset of a sample space and specific collection of outcomes
• Experiment probability- determined based on the results of an experiment repeated many times
• Probability Mass Function (pmf)- probability distribution of a discrete random variable
• Random variable- result of chance event
• Discrete Random Variable- can take only finite (countable) number, values are exact and can be
represented by non-negative numbers

Properties of Discrete Random Variable

1. 0 ≤ P(x) ≤ 1 - Probability of each value of the discrete random variable is between 0 to 1
2. ∑ P(x) = 1 - summation of all the probabilities is equal to 1.

• Continuous Random Variable- can assume an infinite number of values and can be represented by
fraction, decimal and negative numbers

Properties of Probability Mass Function f(x).

a. f(x) = P(X = x) ≥ 0 if x ∈ of the Support - For every element x in the Support S, all the
probabilities should be a positive value.
b. ∑ f(x) = ∑ P(X = x) = 1 - The sum of all the probabilities for all possible x values in the support S
must be equal to 1.

WEEK 2

• Mean - average of a data set, found by adding all numbers together and then dividing the sum of the
numbers by the number of numbers
• Variance - statistical measurement of the spread between numbers in a data set, average of the
squared differences from the mean
• Standard Deviation - dispersion of a dataset relative to its mean and is calculated as the square root
of the variance
• Expected value - discrete random variable X, symbolized as E(X), often referred to as the
long-term average or mean (symbolized as μ)

FORMULA:

Expected Value
𝐸(𝑋) = ∑ 𝑥𝑓(𝑥)
𝑥ϵ𝑆

Mean / Expected Value μ = ∑ xP(x)

Variance 2 2
σ = ∑[(x − μ) P(x)]

Standard Deviation 2
σ= ∑[(𝑥 − μ) 𝑃(𝑥)]

WEEK 3

• Normal Distribution – also known as the Gaussian distribution, bell-shaped curve and type of data
distribution that is observed
• Standard Normal Distribution – normal distribution, mean of 0 and a standard deviation of 1
• Standard normal distribution table – compilation of areas from the standard normal distribution
• Empirical Rule - also called the three-sigma or 68-95-99.7 rule, is a statistical rule which data will
fall within three standard deviations (denoted by the Greek letter sigma, or σ) of the mean or average
(represented by the Greek letter mu, or μ)

WEEK 4

• Population – data set contains all members of specified group

• Sample – subset of a population
• Parameter – all the data values in the population
• Statistic – using only the data values in a sample
• Sampling Distribution – probability distribution for the values of the sample statistic obtained
when random samples are repeatedly drawn from a population

• Random sampling - selection of n elements derived from the N population, sample has an equal
chance of being selected

Random Sampling Technique

Probability Sampling
Every member of the population has the chance of being selected

1. Simple Random Sample- every member of the population has an equal chance of being selected.
Your sampling frame should include the whole population
2. Systematic sampling- slightly easier to conduct. Every member of the population is listed with a
number, individuals are chosen at regular intervals.
3. Stratified sampling- involves dividing the population into subpopulations
4. Cluster sampling- dividing the population into subgroups, but each subgroup should have similar
characteristics to the whole sample.

Non-probability Sampling
Based on non-random criteria, and not every individual has a chance of being included

1. Convenience sampling- individuals who happen to be most accessible to the researcher

2. Voluntary response sampling- based on ease of access, people volunteer themselves (e.g. by
responding to a public online survey). Voluntary response samples are always at least somewhat
biased
3. Purposive Sampling - also known as judgement sampling, involves selecting a sample that is most
useful to the purposes of the research.
4. Snowball Sampling - If the population is hard to access, it can be used to recruit participants via
other participants. The downside here is also representativeness, as you have no way of
knowing how representative your sample is due to the reliance on participants recruiting others. This
can lead to sampling bias.
5. Quota sampling- predetermined number or proportion of units called a quota then divide the
population into mutually exclusive subgroups (called strata) and then recruit sample units. The aim of
quota sampling is to control what or who makes up your sample.

FORMULA

Population mean Σ𝑥
μ=
𝑁

Population Variance 2 Σ(𝑥−µ)

2
σ= 𝑁

Population Standard Deviation 2

σ= σ

Mean of Sampling Distribution Σ𝑥

µ𝑋 = 𝑁

Variance of Sampling Distribution 2 Σ(𝑥−µ𝑥)

2

σ 𝑋
= 𝑁
 Standard deviaiton of Sampling Distribution 2
σ𝑥 = σ𝑥

WEEK 5

• Central Limit Theorem- sampling distribution of the mean approaches a normal distribution, as
the sample size increases

Note 1: shaded area is from the given z-score up to the left side entirely, the located area in the
table is the answer
Note 2: shaded area is from the given z-score up to the right-side entirely, the located area in the
table should be subtracted to 1. (absolute value)
Note 3: shaded area is from the given z-score up to the mean/center, the located area in the table
should be subtracted to 0.5. (absolute value)
Note 4: shaded area is in between two given z-scores, the two located areas in the table should be
subtracted by each other. (absolute value)

Steps in solving problem related to Central Limit Theorem.

1. Given and Find

2. Compute for Z-score using the formula for Z under the Central Limit Theorem
3. Draw a Normal Curve and its shaded area.
4. Locate the corresponding area of the z-score in the z-table then consider note 1,2,3 and 4 with
step 3.
5. Final answer can be in the form of decimal and percentage (two decimal places)

FORMULA

Central Limit theorem 𝑥− µ

Z= σ
𝑛

WEEK 6

• Student’s T-Distribution – used to calculate population parameters when the sample size is small
and population variance is unknown, sample size is less than 30
• William Sealy Gosset- publish t-distribution studies in 1908 under the name “student” that’s why it
is called student’s t-distribution
• Percentile - describes how a score compares to other scores from the same set
• Confidence interval – range of values that is used to estimate a parameter
• Narrowness of the interval – confidence interval's narrow width in comparison to its length
• Degree of freedom - maximum number of logically independent values which vary in the data
sample

FORMULA

T- Table 𝑥− µ
t= 𝑠
𝑛

EBM 𝑠
𝑡 α
𝑑𝑓 2 𝑛

df ( degree of freedom) n − 1 and α

Interval (x̅ − EBM, x̅ + EBM)

Length of the Confidence Level σ σ

x̅- 𝑍 α < 𝑥̄ < 𝑥̄ + 𝑍 α
2 𝑛 2 𝑛

Lower Limit LL = x̄− E

Upper Limit UL = x̄+ E

Minimum Sample Size in Estimating the 𝑍α ∗σ

Population Mean 2
n= ( 2

𝐸
)

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