Source: STAT 211 Engineering Statistics, Texas A&M University Difficulty: Introductory | Prerequisites: Elementary integral calculus
Tags: descriptive statistics, probability, histograms, standard deviation, variance, mean, median, mode, frequency tables, engineering statistics, STAT 211, Texas A&M, FE exam statistics
This is the foundation of the entire course. Before you can run any statistical test or build any model, you need to know how to describe data, visualise it, and reason about chance. These first four weeks cover the vocabulary, the plots, and the probability rules that everything else builds on. If you skipped ahead to confidence intervals or hypothesis testing without this material, most of what you read would not make sense.
Descriptive statistics compress raw data into useful summaries: centres, spreads, and shapes. Probability gives you a formal framework for quantifying uncertainty, which is the entire point of statistics. The normal distribution is the single most important probability distribution in the course, and you will use it in nearly every topic from Week 5 onward.
Population
The complete set of all individuals, items, or measurements of interest. In simple terms, it is everybody or everything you want to draw conclusions about, even if you cannot measure them all.
Sample
A subset of the population that you actually collect data from. Think of it as the slice of reality you can afford to observe.
Parameter
A numerical summary of a population (e.g. the true population mean, μ). You rarely know this exactly; the whole point of statistics is to estimate it.
Statistic
A numerical summary computed from a sample (e.g. the sample mean, x̄). This is what you calculate from the data you have in hand.
Mean (arithmetic mean)
The sum of all observations divided by the number of observations. For a sample: x̄ = (Σxᵢ) / n. In simple terms, it is the balancing point of the data.
Median
The middle value when data are sorted from smallest to largest. If n is even, it is the average of the two middle values. Think of it as the value that splits the data in half.
Mode
The value that occurs most frequently. A dataset can have more than one mode or no mode at all.
Range
The difference between the maximum and minimum values. It is the crudest measure of spread, highly sensitive to outliers.
Variance
The average of the squared deviations from the mean. For a sample: s² = Σ(xᵢ , x̄)² / (n , 1). It measures how spread out the data are, in squared units.
Standard deviation
The square root of the variance: s = √s². In simple terms, it puts the spread back into the original units of measurement, which makes it far more interpretable than variance.
Interquartile range (IQR)
Q3 minus Q1, where Q1 and Q3 are the 25th and 75th percentiles. It captures the spread of the middle 50% of the data and is resistant to outliers.
Histogram
A bar chart that groups continuous data into bins and shows the frequency (or relative frequency) of observations in each bin. The shape tells you about symmetry, skewness, and potential outliers.
Boxplot (box-and-whisker plot)
A visual summary showing the median, Q1, Q3, and potential outliers. Useful for comparing distributions side by side.
Skewness
A measure of asymmetry. Right-skewed (positive skew) means the tail stretches to the right; left-skewed (negative skew) means it stretches to the left. In simple terms, skewness tells you which direction the outliers pull.
Probability
A number between 0 and 1 (inclusive) that represents the likelihood of an event occurring. 0 means impossible, 1 means certain.
Sample space (S)
The set of all possible outcomes of an experiment. Think of it as the complete menu of things that could happen.
Event
A subset of the sample space. Any collection of outcomes you are interested in.
Complement
The complement of event A, written A', is everything in the sample space that is not in A. P(A') = 1 , P(A).
Union
A ∪ B is the event that A or B (or both) occur.
Intersection
A ∩ B is the event that both A and B occur.
Mutually exclusive (disjoint)
Two events that cannot happen at the same time: P(A ∩ B) = 0. In simple terms, if one happens, the other is ruled out.
Independent events
Two events where the occurrence of one does not change the probability of the other: P(A ∩ B) = P(A) · P(B). Do not confuse this with mutually exclusive.
Conditional probability
The probability of A given that B has occurred: P(A|B) = P(A ∩ B) / P(B). In simple terms, you are zooming into the world where B already happened and asking how likely A is within that smaller world.
Normal distribution
A continuous, symmetric, bell-shaped distribution fully described by its mean (μ) and standard deviation (σ). In simple terms, it is the bell curve, and it shows up everywhere in nature and engineering.
Standard normal distribution
A normal distribution with μ = 0 and σ = 1. Any normal random variable X can be converted to a standard normal Z using the Z-score formula.
Z-score
Z = (X , μ) / σ. It tells you how many standard deviations a value is from the mean. Positive means above the mean; negative means below.
Data can be categorical (labels, groups) or quantitative (numerical values that you can do arithmetic on).
Quantitative data is further classified as discrete (countable values, e.g. number of defects) or continuous (measured on a scale, e.g. temperature, length).
A frequency table tallies how often each value or category appears. A relative frequency table expresses those counts as proportions of the total.
Histograms are for continuous data; bar charts are for categorical data. The key visual difference: histogram bars touch (continuous scale), bar chart bars have gaps.
The shape of a histogram matters: symmetric, right-skewed, left-skewed, uniform, or bimodal. Shape influences which summary statistics and which methods are appropriate later.
The mean is sensitive to outliers. A single extreme value can drag it substantially.
The median is resistant to outliers, making it a better measure of centre for skewed data.
For symmetric data, the mean and median are roughly equal.
Standard deviation is the workhorse measure of spread in this course. Nearly every formula from Week 7 onward involves it.
When dividing by (n , 1) instead of n for the sample variance, you are applying Bessel's correction, which compensates for the fact that a sample tends to underestimate population variability.
Addition rule: P(A ∪ B) = P(A) + P(B) , P(A ∩ B). For mutually exclusive events, the last term drops to zero.
Multiplication rule: P(A ∩ B) = P(A) · P(B|A). For independent events, this simplifies to P(A) · P(B).
Complement rule: P(A') = 1 , P(A). Often the fastest route: if the direct calculation is hard, compute the complement instead.
Conditional probability is the foundation for Bayesian thinking and appears repeatedly in later hypothesis-testing topics.
A contingency table (two-way table) cross-tabulates two categorical variables.
Joint probability: the probability of both row and column events, read from the interior cells divided by the grand total.
Marginal probability: the probability of a single variable's event, read from the row or column totals divided by the grand total.
Conditional probability from a table: restrict your attention to one row (or column) total as the new denominator.
The curve is defined by two parameters: μ (centre) and σ (spread).
Approximately 68% of values fall within 1σ of the mean, 95% within 2σ, and 99.7% within 3σ. This is the empirical rule (68-95-99.7 rule).
To find probabilities for any normal random variable, convert to a Z-score and use a standard normal table or software.
Finding X from a probability: work backwards. Look up the Z-value for the given cumulative probability, then solve X = μ + Zσ.
Sample mean: x̄ = (1/n) Σ xᵢ
Sample variance: s² = Σ(xᵢ , x̄)² / (n , 1)
Sample standard deviation: s = √s²
Z-score: Z = (X , μ) / σ
Addition rule: P(A ∪ B) = P(A) + P(B) , P(A ∩ B)
Conditional probability: P(A|B) = P(A ∩ B) / P(B)
Multiplication rule (general): P(A ∩ B) = P(A) · P(B|A)
Multiplication rule (independent): P(A ∩ B) = P(A) · P(B)
Standard deviation is how engineers set quality tolerances in manufacturing: a machine that produces bolts with a small σ is far more reliable than one with a large σ, even if both have the same mean diameter. The normal distribution is behind quality control charts, the design of experiments, and the statistical methods on the FE exam.
Students often confuse mutually exclusive with independent. They are not the same thing, and in fact, two events with non-zero probabilities cannot be both mutually exclusive and independent.
Dividing by n instead of (n , 1) when computing the sample variance is a frequent calculator error. The sample variance formula uses (n , 1).
A common mistake is treating a histogram like a bar chart and leaving gaps between the bars, or vice versa. Histograms have no gaps because the data is continuous.
Students sometimes assume that a large standard deviation means the data is "bad." It simply means there is more variability. Whether that matters depends on context.
⚠️ Z-score calculations appear on nearly every exam. Be comfortable converting in both directions (X to Z and Z to X).
⚠️ The empirical rule (68-95-99.7) is a quick-check tool that examiners use to test conceptual understanding.
⚠️ Conditional probability from a two-way table is a staple exam question. Practise reading tables quickly.
⚠️ Know when to use P(A') = 1 , P(A). "At least one" problems almost always call for the complement approach.
True or False: The median is always equal to the mean for any dataset.
Fill in the blank: The standard normal distribution has a mean of ___ and a standard deviation of ___.
True or False: If P(A) = 0.3 and P(B) = 0.4, and A and B are mutually exclusive, then P(A ∩ B) = 0.12.
Fill in the blank: Bessel's correction divides by ___ instead of n.
True or False: A Z-score of ,2.0 means the observation is two standard deviations below the mean.
Answers: 1. False (only for symmetric distributions). 2. 0, 1. 3. False (mutually exclusive means P(A ∩ B) = 0, not P(A)·P(B)). 4. n , 1. 5. True.
Q: A dataset has values {3, 7, 7, 10, 13}. What are the mean and median?
A: Mean = (3 + 7 + 7 + 10 + 13) / 5 = 40 / 5 = 8. Median = 7 (the middle value when sorted).
Q: Events A and B are independent with P(A) = 0.5 and P(B) = 0.3. What is P(A ∪ B)?
A: P(A ∪ B) = 0.5 + 0.3 , (0.5)(0.3) = 0.8 , 0.15 = 0.65.
Q: A value X comes from a normal distribution with μ = 100 and σ = 15. What is the Z-score for X = 130?
A: Z = (130 , 100) / 15 = 30 / 15 = 2.0. The value is two standard deviations above the mean.
Q: In a contingency table, 40 out of 200 total observations fall in the cell for "Defective" and "Machine A." The marginal total for Machine A is 80. What is the probability of Defective given Machine A?
A: P(Defective | Machine A) = 40 / 80 = 0.50.
Q: What proportion of data falls within one standard deviation of the mean for a normal distribution?
A: Approximately 68%, per the empirical rule.
This material feeds directly into sampling distributions (Week 7), because the Central Limit Theorem relies on the normal distribution and on your understanding of means and standard deviations. Probability rules reappear in hypothesis testing (Weeks 10, 11), where you compute p-values that are, at their core, conditional probabilities.
descriptive statistics, measures of central tendency, measures of spread, dispersion, variability, frequency distribution, relative frequency, cumulative frequency, percentile, quartile, five-number summary, bell curve, Gaussian distribution, standard normal table, Z-table, probability rules, addition rule, multiplication rule, complement rule, conditional probability, joint probability, marginal probability, contingency table, two-way table, empirical rule, 68-95-99.7 rule, Bessel's correction, sample vs population, parameter vs statistic, STAT 211, engineering statistics, FE exam