Course: PSY 301 Understanding Statistics, East Texas A&M Source: Moore & Notz, Statistics: Concepts and Controversies, 10th ed., Chapters 13, 17, 18 Covers: Exam 3 material
Difficulty: Intermediate Prerequisites: Chapters 1, 3, 8 (variables, samples, measurement) and Chapters 10, 11, 12 (graphs, mean, standard deviation). You must be comfortable calculating the mean and standard deviation before tackling z-scores.
Chapter 13 introduces the normal distribution, the single most important shape in statistics. Many real-world measurements (heights, test scores, repeated lab measurements) approximate this bell-shaped curve, and the entire inferential framework you will learn later depends on it. Chapters 17 and 18 then introduce probability, the mathematical language that makes it possible to move from "what did we observe in our sample?" to "what can we say about the population?" Without probability, there is no bridge between data and inference.
The normal distribution is a symmetric, bell-shaped curve defined entirely by its mean and standard deviation. You can use z-scores and a standard normal table to find the percentage of observations in any region. Probability is the long-run relative frequency of an event, governed by a few basic rules (probabilities add to 1, complement rule, addition rule for disjoint events, multiplication rule for independent events). These ideas underpin every inferential method in the rest of the course.
Normal distribution (normal curve, Gaussian distribution)
A symmetric, bell-shaped distribution completely specified by its mean (μ) and standard deviation (σ). Think of it as: the classic bell curve. The mean marks the centre, the standard deviation controls how wide or narrow the bell is.
Standard normal distribution
The normal distribution with mean 0 and standard deviation 1. In simple terms, this is the "reference" version of the normal curve. You convert any normal distribution to this one using z-scores so you can use a single table.
Z-score (standardised score)
The number of standard deviations an observation falls above or below the mean. In simple terms, a z-score tells you where a value sits relative to the rest of the data. Positive means above the mean, negative means below, zero means right at the mean.
68-95-99.7 rule (empirical rule)
In any normal distribution, approximately 68% of values fall within 1 standard deviation of the mean, 95% within 2, and 99.7% within 3. Think of it as: the quick-reference guide for normal distributions. If you know the mean and SD, you know where most of the data sit without looking anything up.
Percentile
The percentage of values in a distribution that fall at or below a given value. Think of it as: scoring at the 90th percentile means you did as well as or better than 90% of the group.
Probability
The long-run proportion of times an event occurs in many repeated trials. In simple terms, probability is not about predicting a single outcome; it describes patterns that emerge over thousands of repetitions.
Sample space
The set of all possible outcomes of a random phenomenon. Think of it as: the complete list of everything that could happen.
Event
A collection of outcomes from the sample space. In simple terms, an event is any subset of possible results that you are interested in.
Complement of an event
Everything in the sample space that is not part of the event. Think of it as: the opposite. If event A is "rolling a 6," the complement of A is "rolling anything other than a 6."
Disjoint events (mutually exclusive events)
Events that cannot happen at the same time. In simple terms, if one occurs, the other cannot. Drawing a heart and drawing a spade from a single card draw are disjoint.
Independent events
Events where the occurrence of one does not change the probability of the other. Think of it as: knowing the result of one event tells you nothing about the other. Successive fair coin flips are independent.
Law of large numbers
As the number of trials increases, the observed proportion of an event gets closer and closer to its true probability. In simple terms, flip a coin 10 times and you might get 70% heads; flip it 10,000 times and you will be very close to 50%.
Personal probability (subjective probability)
A probability that reflects an individual's degree of belief rather than long-run frequency. Think of it as: your gut estimate. "I think there is a 30% chance it will rain" is personal probability. It does not come from repeating the same day thousands of times.
The normal distribution is defined by two parameters:
Mean (μ): the centre of the curve.
Standard deviation (σ): the spread. Larger σ produces a wider, flatter curve; smaller σ produces a taller, narrower one.
Properties of the normal curve:
Perfectly symmetric about the mean.
The total area under the curve equals 1 (representing 100% of observations).
The tails extend infinitely in both directions but never touch the horizontal axis.
The mean, median, and mode are all the same value.
The 68-95-99.7 rule is your primary tool for quick calculations:
68% of data between μ − 1σ and μ + 1σ
95% of data between μ − 2σ and μ + 2σ
99.7% of data between μ − 3σ and μ + 3σ
Z-scores standardise any normal distribution:
z = (x − μ) / σ
After converting, use the standard normal table (Table A in the textbook) to find the area to the left of any z-score. That area equals the proportion of observations at or below that value.
To find the area to the right, subtract the table value from 1.
To find the area between two z-scores, look up both and subtract the smaller area from the larger.
Working problems with the normal distribution follows a consistent pattern:
State the variable and its distribution (mean, SD).
Sketch the curve and shade the region of interest.
Convert the boundary values to z-scores.
Use the table to find the required area.
What probability means:
Probability is a number between 0 and 1 (or 0% and 100%).
A probability of 0 means the event never occurs in the long run. A probability of 1 means it always occurs.
The probability of an event is its long-run relative frequency, established by observing many, many trials.
Basic probability rules:
Rule 1: Any probability is a number between 0 and 1.
Rule 2: The probabilities of all possible outcomes in the sample space must add to 1.
Rule 3 (Complement rule): P(not A) = 1 − P(A). The probability something does not happen equals 1 minus the probability it does.
Rule 4 (Addition rule for disjoint events): If A and B cannot occur together, P(A or B) = P(A) + P(B). You can only add probabilities directly when the events are disjoint.
Rule 5 (Multiplication rule for independent events): If A and B are independent, P(A and B) = P(A) × P(B). You can only multiply probabilities directly when the events are independent.
Common pitfalls with probability:
Do not add probabilities for events that are not disjoint. If events overlap, adding double-counts the shared outcomes.
Do not multiply probabilities for events that are not independent. If one event changes the likelihood of another, simple multiplication gives the wrong answer.
Law of large numbers:
In the short run, randomness is unpredictable. A coin can land heads five times in a row.
In the long run, regularity emerges. Over thousands of flips, the proportion of heads approaches 0.5.
The law does not say that outcomes "balance out." After a run of heads, the coin does not become more likely to land tails. Each flip is independent.
Personal probability is useful in everyday life (estimating the chance your car breaks down, for instance) but is not based on long-run frequency. In formal statistical inference, we rely on frequency-based probability.
Z-score formula:
z = (x − μ) / σ
Where x is the observed value, μ is the population mean, and σ is the population standard deviation.
Solving for x when you know the z-score:
x = μ + z × σ
Complement rule:
P(not A) = 1 − P(A)
Addition rule (disjoint events):
P(A or B) = P(A) + P(B)
Multiplication rule (independent events):
P(A and B) = P(A) × P(B)
Standardised test scores (SAT, GRE, IQ) are designed to follow a normal distribution. When you hear that someone scored "two standard deviations above the mean," that places them in roughly the top 2.5% of test-takers, a direct application of the 68-95-99.7 rule.
Insurance companies rely entirely on probability. They cannot predict whether a specific policyholder will file a claim, but over millions of customers the long-run proportions are stable enough to set premiums. This is the law of large numbers at work.
Students often believe the 68-95-99.7 rule applies to any dataset. It applies only to data that are approximately normally distributed. If the distribution is heavily skewed, the rule does not hold.
Students confuse "disjoint" with "independent." Disjoint events cannot happen simultaneously; independent events can happen simultaneously but do not influence each other. In fact, if two events both have positive probability and are disjoint, they cannot be independent.
Students think that after several heads in a row, tails becomes "due." This is the gambler's fallacy. Each flip of a fair coin is independent; past results do not influence future ones.
Students sometimes forget to sketch the curve and shade the region when solving normal distribution problems. Skipping the sketch is the most common source of errors (for instance, finding the area to the left when the question asks for the area to the right).
⚠️ The z-score formula will appear on the exam. Know how to calculate z and how to reverse the formula to find x.
⚠️ Be prepared to use the 68-95-99.7 rule for quick estimates. Questions may ask "what percentage of values fall between..." without requiring a table.
⚠️ Expect questions that give a normal distribution scenario and ask you to find a proportion using the standard normal table.
⚠️ Know the four basic probability rules and when each applies. The addition rule requires disjoint events; the multiplication rule requires independence.
⚠️ The complement rule is often the fastest path to an answer. If a question asks "what is the probability of at least one," think 1 minus the probability of none.
⚠️ Be able to distinguish between disjoint and independent. This is a common exam trap.
⚠️ All exams are cumulative. Expect questions on variables, sampling, and descriptive statistics alongside this material.
Fill in the blank: In a normal distribution, approximately __________% of values fall within one standard deviation of the mean.
True or false: A z-score of negative 1.5 means the observation is 1.5 standard deviations below the mean.
True or false: Two disjoint events are always independent.
Fill in the blank: The probability of an event and the probability of its complement always sum to __________.
True or false: The law of large numbers says that outcomes must "even out" in the short run.
Answers: 1. 68. 2. True. 3. False. 4. 1. 5. False (it applies to the long run, and outcomes do not "even out" by compensating).
Q: IQ scores are normally distributed with a mean of 100 and a standard deviation of 15. What is the z-score for an IQ of 130?
A: z = (130 − 100) / 15 = 30 / 15 = 2.0. An IQ of 130 is exactly 2 standard deviations above the mean.
Q: Using the 68-95-99.7 rule, what percentage of people have an IQ between 85 and 115?
A: 85 is one standard deviation below the mean and 115 is one standard deviation above. Approximately 68% of people fall in this range.
Q: The probability of rain on any given day in a certain city is 0.30. What is the probability it does not rain?
A: P(not rain) = 1 − 0.30 = 0.70, using the complement rule.
Q: You roll two fair six-sided dice. What is the probability of getting a 6 on both rolls?
A: The rolls are independent, so P(6 and 6) = (1/6) × (1/6) = 1/36 ≈ 0.028.
Q: Events A and B are disjoint. P(A) = 0.3 and P(B) = 0.4. What is P(A or B)? Could A and B also be independent? Explain.
A: P(A or B) = 0.3 + 0.4 = 0.7 (addition rule for disjoint events). A and B cannot be independent. If they were independent, P(A and B) would equal 0.3 × 0.4 = 0.12, but disjoint events have P(A and B) = 0, which contradicts independence.
The normal distribution is the foundation for confidence intervals (Chapter 21) and statistical significance (Chapter 22). The margin of error in a confidence interval depends on the standard deviation and the properties of the normal curve. Probability rules underpin the logic of hypothesis testing: you compute the probability of observing your data under a null hypothesis, and that probability determines whether your result is "statistically significant."
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