STAT101 - Introduction to Probability Theory and Statistics
The biggest, most formula-heavy chapter in the course, rebuilt as one running example: mean, median, mode, quartiles, variance, standard deviation, z-scores, Chebyshev, and the Empirical Rule — all traced through the same seven numbers.
Chapter 2 showed you how to see data (tables, histograms). Chapter 3 shows you how to summarize it with a handful of numbers: where is the center, and how spread out is the data around that center? Almost every exam question in this course reduces to "compute this number correctly" — and almost every point lost on this chapter comes from one of two things: (1) dividing by the wrong denominator (N vs. n−1), or (2) rounding the percentile/median index rule backwards. Get those two habits right and this chapter is mostly arithmetic.
To show how every formula in this chapter connects, we will reuse one small dataset the whole way through: a sample of starting salaries (in dollars) for n = 7 recent business graduates (adapted from the textbook's accounting-graduate salary example):
Because this is a sample (only 7 of all possible graduates), we will use sample notation (x̄, s, s², n) throughout — watch how that choice ripples into every formula below.
| Term | Computed From | Behavior |
|---|---|---|
| Parameter | The entire population | Fixed — does not change unless the population itself changes |
| Statistic | A sample drawn from the population | Varies from sample to sample |
Every measure in this chapter has two versions: a population version (Greek letters: μ, σ, σ²) and a sample version (Roman letters: x̄, s, s²). The formula is the same idea but the notation — and one key denominator — changes.
The mean is the arithmetic average — sum all the values and divide by how many there are. It is the "balance point" of the data: if you placed each data value as a weight on a number line, the mean is where the line would balance.
μ = population mean | N = population size | x = individual data value
x̄ = sample mean | n = sample size
Data (n = 7): 39,000, 44,000, 46,000, 52,000, 56,000, 60,000, 61,000
Interpretation: the average starting salary in this sample of 7 graduates is about
$51,143. Excel: =AVERAGE(range).
Before you divide anything, ask: "Do I have data on every single member of the group I care about, or just some of them?"
This one question resolves 90% of "which formula do I use" confusion in this chapter.
The sum of the deviations of every data value from the mean always equals zero: Σ(x − μ) = 0. This is why the mean is called the "balance point" of the data — positive deviations above the mean exactly cancel negative deviations below it.
The mean uses every data value, which is its strength — and its weakness. It is very sensitive to extreme values (outliers). Classic textbook example: 7 accounting graduates' salaries average $51,143. Add one outlier — a graduate who was a 1st-round NFL draft pick earning $2,300,000 — and the mean of the (now n = 8) group explodes to $332,250, even though 7 of the 8 people still earn about $50,000. One extreme value can make the mean meaningless as a "typical" value. This is exactly why the median exists.
The median (Md) is the middle value of a data array (data sorted from lowest to highest). It splits the distribution exactly in half: 50% of values fall at or below it, 50% at or above it. Unlike the mean, it is not affected by extreme values.
Step 1: Collect the data. Step 2: Sort it from smallest to largest. Step 3: Compute the median index i = (n+1)/2. Step 4: Find the median.
Sorted data (n = 7, odd): 39,000, 44,000, 46,000, 52,000, 56,000, 60,000, 61,000
Compare: mean = $51,143 vs. median = $52,000 — very close, which tells us this small dataset is fairly symmetric with no extreme values pulling the mean away from the center (confirmed later by the outlier check in Section 3.3).
If n were even (e.g., the classic "number of sales" example: 5, 10, 15, 20, 20, 20, 30, 40, 50, 90 with n = 10), the median is the average of the 5th and 6th sorted values: (20 + 20)/2 = 20.
The general median-index rule (also used for percentiles below) is: compute i. If i is not an integer, round UP to the next highest integer and take that position's value. If i IS an integer, the median is the average of the values at position i and i+1. Students often round to the "nearest" integer instead of always rounding up — that gives the wrong position when i has a decimal like .5 that would round down.
The mode is the value that occurs most frequently in the data. It is the only measure of center that works on any level of data, including nominal (category) data. A dataset can have no mode, one mode (unimodal), two modes (bimodal), or more.
Data: 39,000, 44,000, 46,000, 52,000, 56,000, 60,000, 61,000
Every value appears exactly once → this dataset has no mode. (Compare to the classic sales-data example 5,10,15,20,20,20,30,40,50,90 where 20 appears three times, so Mode = 20.)
The weighted mean is used when some values matter more than others (they carry different "weights" — e.g., different numbers of shares, different class sizes, different quantities purchased).
w = the weight (relative importance) assigned to each value x
| Mutual Fund | Return, x (%) | Shares, w | w × x |
|---|---|---|---|
| Fidelity | 7.2 | 2,000 | 14,400 |
| Vanguard | 8.3 | 5,000 | 41,500 |
| Dimensional | 5.4 | 12,000 | 64,800 |
| Total | 19,000 | 120,700 |
Note the plain average of 7.2, 8.3, and 5.4 would be 6.97% — noticeably different, because the weighted mean correctly gives Dimensional's 5.4% return more influence (it has the most shares).
The p-th percentile is the value below which p% of the sorted data falls (and above which (100−p)% falls). Quartiles are just specific, commonly-used percentiles:
| Quartile | Equivalent Percentile |
|---|---|
| Q1 (1st quartile) | 25th percentile |
| Q2 (2nd quartile) | 50th percentile = the median |
| Q3 (3rd quartile) | 75th percentile |
Step 1: sort the data lowest to highest. Step 2: compute i. Step 3: if i is not an integer, round UP to the next integer — that position's value is the p-th percentile. If i IS an integer, the p-th percentile is the average of the values at position i and i+1.
Sorted data (miles): 4.1, 4.4, 5.8, 6.5, 8.6, 9.6, 10.1, 11.1, 11.5, 11.7, 12.2, 12.4, 13.0, 13.1, 13.2, 13.4, 13.5, 13.8, 14.1, 14.6, 15.7, 16.2, 19.3, 20.5, 21.0, 21.4, 21.7, 23.7, 24.9, 26.9
Because i = 24 is an integer, the 80th percentile is the average of the 24th and 25th sorted values: (20.5 + 21.4)/2 = 20.95 miles. (The textbook's own worked answer of 20.75 uses the values 20.5 and 21.0 for positions 24 and 25 — always double-check your own sort order against the printed data table.)
Interpretation: 80% of moving jobs in this sample traveled about 21 miles or less.
Sorted data (n = 7): 39,000, 44,000, 46,000, 52,000, 56,000, 60,000, 61,000
A graph built from the five-number summary: minimum, Q1, median, Q3, maximum.
Every one of our 7 salaries (39,000–61,000) falls comfortably inside [$20,000, $84,000] → no outliers by the box-plot rule. We will confirm this again with z-scores in Section 3.3.
| Shape | Relationship of Mean, Median, Mode | What Pulls the Mean |
|---|---|---|
| Symmetric (bell-shaped) | Mean = Median = Mode (all at the center) | Nothing — balanced on both sides |
| Left-skewed (negative skewness) | Mean < Median < Mode | A long tail of low extreme values pulls the mean down/left |
| Right-skewed (positive skewness) | Mode < Median < Mean | A long tail of high extreme values pulls the mean up/right |
Just compare mean and median:
In our salary example, mean ($51,143) is just barely below median ($52,000) — essentially symmetric, consistent with having no outliers.
Excel and calculators will happily compute a mean for any numbers you give them — even if those numbers are just category codes. Example: Marital Status coded 1 = Single, 2 = Married, 3 = Divorced, 4 = Widowed. A "mean" of 1.93 does not mean "almost married" — the numbers are just labels, and averaging labels is meaningless. Same caution applies to 5-point Likert scale survey data (Strongly Disagree=1 ... Strongly Agree=5): the mean assumes equal spacing between categories, which is often not a safe assumption.
| Measure | Ratio / Interval | Ordinal | Nominal |
|---|---|---|---|
| Mean | ✓ Appropriate | ✗ Not recommended | ✗ Never |
| Median | ✓ Appropriate | ✓ OK | ✗ Never |
| Mode | ✓ Appropriate | ✓ OK | ✓ Only option for category data |
Two datasets can have the exact same mean and median and still look completely different — because they differ in how spread out the values are. Measures of variation quantify that spread: Range, Interquartile Range, Variance, Standard Deviation, and Coefficient of Variation.
Very easy to compute, but uses only two values (the extremes) and ignores everything in between — and is extremely sensitive to outliers.
Because IQR only looks at the middle 50% of the data, it eliminates the influence of the most extreme high and low values — a more robust measure of spread than the range. From Section 3.1: IQR = $60,000 − $44,000 = $16,000 for our salary data.
Variance is the average of the squared distances of each data value from the mean. We square the deviations because the raw deviations always sum to zero (Section 3.1) — squaring makes every term positive so they don't cancel out.
Population variance divides by N. Sample variance divides by n − 1, NOT n. This is called Bessel's correction, and it exists because a sample's own mean is, by construction, closer to its own data points than the true (unknown) population mean would be — so dividing by n alone would systematically underestimate the true population variance. Dividing by the slightly smaller number n−1 corrects for that bias. Every single time you compute a variance or standard deviation on this exam, the very first thing you must decide is: population (÷ N) or sample (÷ n−1)? Getting this backwards is the single most common point-loss on this chapter.
| Population | Sample | |
|---|---|---|
| Mean | μ = Σx / N | x̄ = Σx / n |
| Variance | σ² = Σ(x−μ)² / N | s² = Σ(x−x̄)² / (n−1) |
| Standard Deviation | σ = √σ² | s = √s² |
x̄ = $51,142.86 (n = 7, so this is a sample → divide by n−1 = 6)
| x | x − x̄ | (x − x̄)² |
|---|---|---|
| 39,000 | −12,142.86 | 147,449,000 |
| 44,000 | −7,142.86 | 51,020,000 |
| 46,000 | −5,142.86 | 26,449,000 |
| 52,000 | 857.14 | 734,700 |
| 56,000 | 4,857.14 | 23,592,000 |
| 60,000 | 8,857.14 | 78,449,000 |
| 61,000 | 9,857.14 | 97,163,000 |
| Sum | ≈ 0 | 424,857,000 |
Matches Excel's =STDEV.S(range) result of $8,415 exactly. Notice the deviation column (x
− x̄) sums to (approximately) zero — the balance-point property from Section 3.1
still holds.
Variance is measured in squared units (e.g., dollars², which is meaningless to interpret directly). Standard deviation is the square root of variance, bringing it back to the original units (dollars) — which is why standard deviation, not variance, is what we actually use to describe "typical distance from the mean" in plain language. If an exam question gives you variance and asks for standard deviation (or vice versa), remember: s = √(s²) and s² = s × s.
These are three different spread measures and are not interchangeable: Range uses only the max and min (very sensitive to outliers). IQR uses only Q1 and Q3 (resistant to outliers, ignores the outer 50% of data). Standard deviation uses every single data point (most information, but still sensitive to extreme values because of the squaring step). Don't substitute one for another on an exam — they answer different questions.
Standard deviation is measured in the original units, which makes it impossible to directly compare variation between two datasets measured in different units (dollars vs. miles) or wildly different scales ($50 stock vs. $100 stock). The coefficient of variation solves this by expressing standard deviation as a percentage of the mean — a unit-free measure of relative variation.
(Population version: CV = (σ / μ) × 100%)
Stock A: average price = $50, standard deviation = $5. Stock B: average price = $100, standard deviation = $5.
Both stocks have the same raw standard deviation ($5), which might tempt you to call them "equally variable." But relative to price, Stock A is twice as variable as Stock B — the CV reveals what the raw standard deviation hides.
Second example (university entrance scores): University A: mean = 70, s = 10 → CV = 10/70(100) = 14.2%. University B: mean = 750, s = 70 → CV = 70/750(100) = 9.3%. Even though University B's raw standard deviation (70) is much bigger than University A's (10), University A actually has more relative variability once you account for the different scales.
CV for our salary data: CV = (8,415 / 51,143) × 100% ≈ 16.5%.
A z-score tells you how many standard deviations a specific data value is above or below the mean. It "standardizes" data onto a common scale, which is what lets you compare values from completely different distributions (different means, different units, different scales).
IQ scores: bell-shaped, μ = 100, σ = 15. A person has an IQ of 121.
Interpretation: this person's IQ is 1.40 standard deviations above the population mean.
University A (100-point scale): μ = 70, σ = 10. University B (1000-point scale): μ = 750, σ = 70. You scored 88 at University A and 798 at University B.
Even though 798 is a "bigger number" than 88, your z-score shows your relative performance was actually much stronger at University A (1.8 standard deviations above average vs. only 0.69).
x̄ = $51,142.86, s = $8,415
Every salary in our dataset has |z| well under 2 — another confirmation that this sample has no outliers.
Chebyshev's Theorem gives a guaranteed minimum percentage of data that falls within k standard deviations of the mean — and remarkably, it works for ANY distribution shape (skewed, bimodal, bell-shaped, anything), as long as k > 1.
| k | Minimum % within k standard deviations |
|---|---|
| k = 2 | 1 − 1/4 = at least 75% |
| k = 3 | 1 − 1/9 = at least 88.9% |
The Empirical Rule is much more specific than Chebyshev's Theorem — but it only applies when the data distribution is bell-shaped (approximately normal). If the data is bell-shaped:
Travel distances are bell-shaped with mean = 15.1 miles, standard deviation = 3.1 miles.
| Chebyshev's Theorem | Empirical Rule | |
|---|---|---|
| Works for which shapes? | ANY distribution shape | Bell-shaped (approximately normal) ONLY |
| Type of statement | Guaranteed minimum ("at least") | Approximate ("approximately") |
| Precision | Conservative / looser bound | More precise, but only if bell-shaped assumption holds |
| k = 2 result | At least 75% | Approximately 95% |
Read the question. If it says or implies "bell-shaped" / "normal" / "symmetric mound-shaped" — use the Empirical Rule for a tighter, more useful answer. If the shape is unknown, skewed, or the question explicitly says "regardless of shape" — use Chebyshev's Theorem. Chebyshev is the safe fallback that always works; the Empirical Rule is the sharper tool you can only use when you know (or are told) the data is bell-shaped.
Two standard methods, both of which you should know:
z-score method: largest |z| in our dataset is 1.44 (for $39,000) — well under 2. No outliers.
1.5 × IQR method (from Section 3.1): Lower Limit = $20,000, Upper Limit = $84,000. All 7 salaries (39,000–61,000) fall inside this range. No outliers.
Both methods agree — this is a well-behaved, roughly symmetric sample with no extreme values, which is exactly why mean and median came out so close together earlier.
If the z-score method and the 1.5×IQR method disagree sharply about whether a value is an outlier, double-check your arithmetic before trusting either answer — on well-behaved data they should generally point to the same conclusion, as they do above.
1. A company has data for ALL of its employees' salaries. Computing the average of this data produces a:
2. Which measure of center is LEAST affected by an extreme outlier?
3. For a SAMPLE of size n, the sample variance formula divides the sum of squared deviations by:
4. A distribution where Mean > Median > Mode is described as:
5. The interquartile range (IQR) is calculated as:
6. Which rule guarantees a minimum percentage of data within k standard deviations of the mean for ANY distribution shape?
7. According to the Empirical Rule, approximately what percentage of bell-shaped data falls within μ ± 2σ?
8. The coefficient of variation is most useful for:
9. A data value with a z-score of 2.9 would generally be considered:
10. When computing the weighted mean, the weight w represents:
11. Variance is expressed in what kind of units?
SA1. A SAMPLE of 5 exam scores is: 70, 75, 80, 85, 90. Compute the sample mean and sample standard deviation.
x̄ = (70+75+80+85+90)/5 = 400/5 = 80
Deviations: −10, −5, 0, 5, 10 → squared: 100, 25, 0, 25, 100 → sum = 250
Because this is a sample (n = 5), divide by n−1 = 4: s² = 250/4 = 62.5
s = √62.5 ≈ 7.91
SA2. Using the running example dataset (starting salaries: 39,000, 44,000, 46,000, 52,000, 56,000, 60,000, 61,000), find Q1, Q3, and the interquartile range. Then determine whether $70,000 would be flagged as an outlier by the 1.5×IQR rule.
n = 7. Q1 index = (25/100)(7) = 1.75 → round up to position 2 → Q1 = $44,000.
Q3 index = (75/100)(7) = 5.25 → round up to position 6 → Q3 = $60,000.
IQR = 60,000 − 44,000 = $16,000.
Upper limit = Q3 + 1.5(IQR) = 60,000 + 24,000 = $84,000.
Since $70,000 < $84,000, it would NOT be flagged as an outlier — it falls inside the upper limit.
SA3. Two machines produce parts with the following stats: Machine X has mean weight 250g, standard deviation 12.5g. Machine Y has mean weight 80g, standard deviation 6g. Which machine has more relative variability? Show your work using the coefficient of variation.
CV(X) = (12.5/250) × 100% = 5%
CV(Y) = (6/80) × 100% = 7.5%
Even though Machine X has a bigger raw standard deviation (12.5g vs. 6g), Machine Y has the higher coefficient of variation, so Machine Y has more relative variability once its much smaller average weight is accounted for.
| Term | Definition |
|---|---|
| Mean | The arithmetic average of a dataset; sum of all values divided by the count. Sensitive to extreme values. |
| Median | The middle value of a sorted (ordered) dataset; splits the data 50/50. Resistant to extreme values. |
| Mode | The value(s) that occur most frequently in a dataset. Works for any level of data, including nominal. |
| Weighted Mean | A mean where each value is multiplied by a weight reflecting its relative importance before averaging: x̄w = Σ(wx)/Σw. |
| Percentile | The value below which a given percentage p of the sorted data falls. |
| Quartile | One of three values (Q1, Q2, Q3) that divide sorted data into four equal parts (25% increments). |
| IQR (Interquartile Range) | Q3 − Q1; the spread of the middle 50% of the data, resistant to outliers. |
| Range | Maximum value minus minimum value; the simplest, most outlier-sensitive measure of spread. |
| Variance | The average of the squared deviations from the mean; population divides by N, sample divides by n−1. Measured in squared units. |
| Standard Deviation | The square root of the variance; brings variability back into the original units of the data. |
| Coefficient of Variation (CV) | Standard deviation expressed as a percentage of the mean; used to compare relative variability across different units or scales: CV = (S/x̄)×100%. |
| Z-score | The number of standard deviations a data value lies above or below the mean; used to standardize and compare values across distributions. |
| Chebyshev's Theorem | For ANY distribution shape, at least (1 − 1/k²) of the data lies within k standard deviations of the mean (k > 1). |
| Empirical Rule | For bell-shaped (approximately normal) distributions only: about 68% of data lies within 1 standard deviation of the mean, 95% within 2, and 99.7% within 3. |
| Skewness | A measure of asymmetry in a distribution's shape; right-skewed has Mean > Median, left-skewed has Mean < Median. |
| Outlier | An extreme data value, typically flagged by |z| > 2 or 3, or by falling outside Q1−1.5(IQR) / Q3+1.5(IQR). |