Introduction to Statistics & Probability

Welcome, Truth Seeker!

Statistics helps us understand and describe data. In a world full of charts, surveys, and studies, knowing statistics helps you evaluate claims and avoid being deceived. We'll learn to analyze data wisely and recognize when statistics are being misused - a vital skill for discerning truth!

The 6Rs Learning Method: RECEIVE, REFLECT, RECALL, RECITE, REVIEW, RESPOND

Lesson 1: Introduction to Statistics

RECEIVE - Learn the Truth

What is Statistics?

Statistics is the science of collecting, organizing, analyzing, and interpreting data to make decisions or draw conclusions.

Key Terms:

Data - Facts or information collected for analysis
Population - The entire group you want to study
Sample - A smaller group selected from the population
Variable - A characteristic that can vary (height, age, opinion)
Statistic - A number that describes a sample
Parameter - A number that describes a population

Types of Data:

Quantitative (Numerical) - Numbers that can be measured or counted
- Discrete - Countable (number of students)
- Continuous - Measurable (height, weight)
Qualitative (Categorical) - Describes qualities or categories (color, gender, opinion)

Population vs. Sample:

You want to know the average height of all 12-year-olds in the US (population = millions). You can't measure them all, so you measure 500 randomly selected 12-year-olds (sample) and use that to estimate the population.

"The first to plead his case seems right, until another comes and examines him." (Proverbs 18:17) Statistics help us examine claims carefully rather than accepting them at face value.

RECALL - Remember the Facts

1 A population is the entire group you want to study.

2 A sample is a smaller group selected from the population.

3 Quantitative data involves numbers; qualitative data describes categories.

4 The number of students in a class is an example of discrete data.

RESPOND - Apply Your Knowledge

5 Why is it usually necessary to use a sample instead of studying the entire population?

Lesson 2: Collecting Data

RECEIVE - Learn the Truth

How Data is Collected

The way data is collected affects its reliability. Poor collection methods lead to unreliable conclusions.

Sampling Methods:

Random Sample - Everyone has an equal chance of being selected (most reliable)
Systematic Sample - Every nth person (e.g., every 10th)
Stratified Sample - Divide population into groups, sample from each
Convenience Sample - Whoever is easiest to reach (least reliable)

Bias - Anything that causes the sample to not represent the population fairly:

Selection Bias - Sample doesn't represent population
Response Bias - Question wording influences answers
Non-response Bias - People who don't respond may differ from those who do

Watch for Bias: If a survey about exercise is conducted at a gym, the results will be biased - gym-goers exercise more than average! Always ask: "Who was surveyed? How were they selected?"

Biased Question: "Don't you agree that we should protect our beautiful parks?"

Neutral Question: "Do you support the proposed park protection plan?"

The first question leads people toward "yes" - this is response bias.

"A false balance is an abomination to Yahuah, but a just weight is His delight." (Proverbs 11:1) Yahuah values honesty in measurements - including data collection!

RECALL - Remember the Facts

1 A random sample gives everyone an equal chance of being selected.

2 Bias causes a sample to not represent the population fairly.

3 Convenience sampling is the least reliable method.

4 Response bias occurs when question wording influences answers.

RESPOND - Apply Your Knowledge

5 You want to survey students about school lunches. Why would surveying only students in the cafeteria be biased?

Lesson 3: Displaying Data

RECEIVE - Learn the Truth

Visual Representations of Data

Graphs and charts help us see patterns in data. Different types work best for different data.

Types of Graphs:

Bar Graph - Compares categories; bars don't touch
Histogram - Shows frequency of numerical ranges; bars touch
Line Graph - Shows change over time
Pie Chart (Circle Graph) - Shows parts of a whole (percentages)
Scatter Plot - Shows relationship between two variables
Box Plot (Box and Whisker) - Shows distribution with median, quartiles

Reading Graphs Carefully:

Check the scale - Does it start at zero?
Look at axis labels - What's being measured?
Check the intervals - Are they consistent?
Note the source - Who created it? Any bias?

Misleading Graphs: A graph can technically show accurate data but still be misleading. Starting the y-axis at 98 instead of 0 can make a small difference look huge. Always check the scale!

"Buy truth and do not sell it; get wisdom, instruction, and understanding." (Proverbs 23:23) Wisdom includes learning to read data correctly!

RECALL - Remember the Facts

1 A histogram shows frequency of numerical ranges with touching bars.

2 A line graph is best for showing change over time.

3 A pie chart shows parts of a whole as percentages.

4 A scatter plot shows the relationship between two variables.

RESPOND - Apply Your Knowledge

5 Which type of graph would you use to show: (a) favorite colors in your class (b) your height over the years (c) how you spend your day

Lesson 4: Mean, Median, Mode

RECEIVE - Learn the Truth

Measures of Central Tendency

These numbers describe the "center" or "typical value" of a data set.

Mean (Average):

Add all values and divide by the number of values.

Mean = Sum of all values ÷ Number of values

Median:

The middle value when data is arranged in order. If even number of values, average the two middle ones.

Mode:

The value that appears most often. There can be no mode, one mode, or multiple modes.

Data Set: 12, 15, 15, 18, 20, 22, 25

Mean: (12+15+15+18+20+22+25) ÷ 7 = 127 ÷ 7 = 18.14

Median: 18 (the middle value)

Mode: 15 (appears twice)

When to Use Each:

Mean - Best for symmetric data without outliers
Median - Best when there are outliers (extreme values)
Mode - Best for categorical data or finding most common value

The Misleading Mean: If 9 people earn $50,000 and 1 person earns $1,000,000, the mean income is $145,000 - but this doesn't represent the typical person! The median ($50,000) is more representative. Always ask which measure is being used!

"Differing weights and differing measures - Yahuah detests them both." (Proverbs 20:10) Using the wrong measure to mislead is a form of dishonesty.

RECALL - Remember the Facts

1 The mean is calculated by adding all values and dividing by the count.

2 The median is the middle value in ordered data.

3 The mode is the most frequent value.

4 The median is best when there are outliers.

REVIEW - Practice Problems

5 Find the mean, median, and mode: 8, 10, 10, 12, 15, 20
Mean: _______ Median: _______ Mode: _______

Lesson 5: Range & Variation

RECEIVE - Learn the Truth

Measures of Spread

These describe how spread out the data is - not just the center, but the variation.

Range:

Range = Maximum value - Minimum value

Quartiles:

Q1 (First Quartile) - 25% of data is below this value
Q2 (Second Quartile) - The median; 50% below
Q3 (Third Quartile) - 75% of data is below this value
IQR (Interquartile Range) - Q3 - Q1 (middle 50% of data)

Data: 10, 12, 14, 15, 18, 20, 22, 25, 30

Range: 30 - 10 = 20

Q1: 13 (median of lower half: 10, 12, 14, 15)

Q2 (Median): 18

Q3: 23.5 (median of upper half: 20, 22, 25, 30)

IQR: 23.5 - 13 = 10.5

Outliers:

Values that are much higher or lower than the rest. Generally, values below Q1 - 1.5(IQR) or above Q3 + 1.5(IQR) are considered outliers.

"A person's wisdom yields patience; it is to one's glory to overlook an offense." (Proverbs 19:11) Understanding variation helps us not overreact to small differences.

RECALL - Remember the Facts

1 Range equals maximum minus minimum.

2 Q2 is the same as the median.

3 IQR = Q3 - Q1.

4 Outliers are values much higher or lower than the rest.

REVIEW - Practice Problems

5 Find the range: 45, 52, 38, 61, 49, 55 Range = _______

Lesson 6: Introduction to Probability

RECEIVE - Learn the Truth

What is Probability?

Probability measures how likely an event is to occur. It's expressed as a number from 0 (impossible) to 1 (certain), or as a percentage (0% to 100%).

Probability = Number of favorable outcomes ÷ Total number of possible outcomes

Key Terms:

Experiment - A procedure with uncertain outcomes (rolling a die)
Outcome - A possible result (rolling a 4)
Sample Space - All possible outcomes (1, 2, 3, 4, 5, 6)
Event - A specific outcome or set of outcomes (rolling an even number)

Example: Rolling a standard die

P(rolling a 3) = 1/6 ≈ 0.167 ≈ 16.7%

P(rolling an even number) = 3/6 = 1/2 = 0.5 = 50%

P(rolling a 7) = 0/6 = 0 = 0% (impossible)

Types of Probability:

Theoretical - Based on mathematical reasoning (a coin has 50% chance of heads)
Experimental - Based on actual experiments (flipping a coin 100 times)

"The lot is cast into the lap, but its every decision is from Yahuah." (Proverbs 16:33) Even events that seem random are under Yahuah's sovereign control!

RECALL - Remember the Facts

1 Probability ranges from 0 (impossible) to 1 (certain).

2 The sample space is all possible outcomes.

3 Theoretical probability is based on reasoning; experimental is based on trials.

REVIEW - Practice Problems

4 A bag has 5 red, 3 blue, and 2 green marbles. P(red) = _______ P(blue) = _______ P(green) = _______

Lesson 7: Probability Calculations

RECEIVE - Learn the Truth

Complement

The complement of an event is everything that is NOT that event.

P(not A) = 1 - P(A)

P(rolling a 6) = 1/6, so P(NOT rolling a 6) = 1 - 1/6 = 5/6

Odds

Odds compare favorable outcomes to unfavorable outcomes (not total outcomes like probability).

Odds in favor = Favorable : Unfavorable
Odds against = Unfavorable : Favorable

Rolling a 6: P = 1/6

Odds in favor = 1:5 (1 way to win, 5 ways to lose)

Odds against = 5:1

Expected Value

The average outcome if an experiment is repeated many times.

Expected Value = Σ(outcome × probability)

Gambling: Casinos and lotteries always have a negative expected value for players - the "house always wins" in the long run. This is why gambling is unwise. "Whoever loves money never has enough" (Ecclesiastes 5:10).

"Dishonest money dwindles away, but whoever gathers money little by little makes it grow." (Proverbs 13:11) Get-rich-quick schemes (including gambling) don't honor Yahuah's way of blessing through honest work.

RECALL - Remember the Facts

1 P(not A) = 1 - P(A).

2 Odds compare favorable to unfavorable outcomes.

3 Expected value is the average outcome over many trials.

REVIEW - Practice Problems

4 If P(rain) = 0.3, what is P(no rain)? _______

5 If 3 out of 10 students are left-handed, what are the odds in favor of randomly selecting a left-handed student? _______

Lesson 8: Combined Events

RECEIVE - Learn the Truth

Independent Events

Events where one outcome doesn't affect the other (flipping a coin twice).

P(A and B) = P(A) × P(B)

P(heads twice in a row) = 1/2 × 1/2 = 1/4 = 25%

Dependent Events

Events where one outcome affects the other (drawing cards without replacement).

P(A and B) = P(A) × P(B given A happened)

Drawing 2 aces from a deck without replacement:

P(first ace) = 4/52

P(second ace | first was ace) = 3/51

P(both aces) = 4/52 × 3/51 = 12/2652 ≈ 0.45%

Addition Rule (Or)

For mutually exclusive events (can't happen together):

P(A or B) = P(A) + P(B)

P(rolling a 2 or a 5) = 1/6 + 1/6 = 2/6 = 1/3

"Trust in Yahuah with all your heart and lean not on your own understanding." (Proverbs 3:5) Understanding probability helps us make wise decisions, but ultimately we trust in Yahuah, not chance.

RECALL - Remember the Facts

1 For independent events, P(A and B) = P(A) × P(B).

2 Events are dependent when one outcome affects the other.

3 For mutually exclusive events, P(A or B) = P(A) + P(B).

REVIEW - Practice Problems

4 A coin is flipped 3 times. P(all heads) = _______

5 A die is rolled. P(1 or 6) = _______

Lesson 9: Misleading Statistics

RECEIVE - Learn the Truth

How Statistics Can Deceive

Statistics can be used to mislead, even without lying. Here are common tricks to watch for:

1. Cherry-Picking Data

Showing only data that supports your position while hiding contradicting data.

2. Misleading Graphs

Y-axis not starting at zero (exaggerates differences)
Unequal intervals
Misleading 3D effects
Truncated axes

3. Confusing Correlation with Causation

Just because two things happen together doesn't mean one causes the other!

Ice cream sales and drownings both increase in summer. Does ice cream cause drowning? No! Both are caused by hot weather.

4. Using the Wrong Average

Choosing mean, median, or mode to make data look better or worse.

5. Small or Biased Samples

"4 out of 5 dentists recommend..." - but how many dentists were asked? How were they selected?

6. Misleading Percentages

"50% increase!" sounds big, but if you went from 2 to 3, that's only 1 more.

Always Ask: Who conducted the study? Who paid for it? What was the sample size? What questions were asked? What data isn't being shown?

"The simple believes every word, but the prudent considers well his steps." (Proverbs 14:15) Don't believe every statistic - think critically!

RECALL - Remember the Facts

1 Cherry-picking means showing only data that supports your position.

2 Correlation does not prove causation.

3 A y-axis not starting at zero can exaggerate differences.

4 "4 out of 5 dentists" is meaningless without knowing the sample size.

RESPOND - Apply Your Knowledge

5 Find an example of a misleading statistic in news or advertising. What makes it misleading?

Lesson 10: Making Wise Decisions

RECEIVE - Learn the Truth

Applying Statistics Wisely

Statistics is a tool for making better decisions, but wisdom comes from Yahuah.

Evaluating Claims:

What is the source? Is it credible?
What was the sample size?
How was the sample selected?
What questions were asked?
What data might be missing?
Does correlation prove causation?
Which "average" is being used?
Are the graphs accurate?

Statistics in Real Life:

Medical Studies - Evaluate sample sizes, funding sources, and conflicts of interest
Polls - Consider margin of error, wording, and sampling method
Advertising - Be skeptical of claims without context
Scientific Claims - Distinguish between observed data and interpretations

Evolution Statistics: Claims about evolution often involve assumptions built into dating methods and interpretation of data. When you hear "scientists say," ask what the actual data shows versus what interpretation is added. The same data can support different conclusions depending on starting assumptions!

Ultimate Wisdom:

Statistics can inform our decisions, but ultimate truth comes from Yahuah's Word. A claim supported by statistics but contradicting Scripture should be questioned. True wisdom begins with fearing Yahuah (Proverbs 9:10).

"The fear of Yahuah is the beginning of wisdom, and the knowledge of the Holy One is understanding." (Proverbs 9:10)

RECALL - Remember the Facts

1 Always consider the source of statistics.

2 The same data can support different conclusions/interpretations depending on assumptions.

3 True wisdom begins with fearing Yahuah.

RESPOND - Apply Your Knowledge

4 Why is it important to think critically about statistics, especially regarding claims about origins?

5 What has been the most important thing you learned in this statistics course?

6 How will you use what you learned to evaluate claims you encounter in the future?

Answer Key

Lesson 1

1. population | 2. sample | 3. Quantitative, qualitative | 4. discrete

Lesson 2

1. random | 2. Bias | 3. Convenience | 4. Response

Lesson 3

1. histogram | 2. line graph | 3. pie chart | 4. scatter plot | 5. (a) bar graph (b) line graph (c) pie chart

Lesson 4

1. mean | 2. median | 3. mode | 4. median | 5. Mean=12.5, Median=11, Mode=10

Lesson 5

1. Range | 2. median | 3. Q3, Q1 | 4. Outliers | 5. 61-38=23

Lesson 6

1. 0, 1 | 2. sample space | 3. Theoretical, experimental | 4. 5/10=1/2, 3/10, 2/10=1/5

Lesson 7

1. 1-P(A) | 2. unfavorable | 3. Expected value | 4. 0.7 | 5. 3:7

Lesson 8

1. P(A)×P(B) | 2. dependent | 3. P(A)+P(B) | 4. 1/2×1/2×1/2=1/8 | 5. 1/6+1/6=2/6=1/3

Lesson 9

1. Cherry-picking | 2. Correlation | 3. zero | 4. sample size

Lesson 10

1. source | 2. conclusions/interpretations | 3. Yahuah