STAT101 - Introduction to Probability Theory and Statistics
The Normal Probability Distribution (Section 6.1)
The normal distribution is the single most important distribution in all of statistics — it's the bell curve behind test scores, heights, measurement errors, and (later in the course) sampling distributions used for confidence intervals and hypothesis tests. This chapter is really just the continuous version of the Empirical Rule you learned in Chapter 3, formalized with a precise tool: the z-score and the standard normal table. Master four skill types — P(Z<z), P(Z>z), P(a<Z<b), and the reverse "find x" problem — and you can answer virtually any exam question in this chapter.
The normal distribution is a continuous, bell-shaped distribution with these key properties:
Changing μ (the mean) shifts the whole curve left or right without changing its shape. Changing σ (the standard deviation) stretches or squeezes the curve — a larger σ means a shorter, wider curve; a smaller σ means a taller, narrower curve.
Remember the Empirical Rule from Chapter 3 for bell-shaped data: about 68% of values fall within 1 standard deviation of the mean, 95% within 2, and 99.7% within 3. That is exactly what the normal distribution formalizes: P(μ−1σ < x < μ+1σ) ≈ 0.6826, P(μ−2σ < x < μ+2σ) ≈ 0.9544, P(μ−3σ < x < μ+3σ) ≈ 0.9974. The z-table just lets you get exact numbers instead of these three rounded approximations.
The standard normal distribution is the one specific normal distribution with mean = 0 and standard deviation = 1. Its horizontal axis is scaled in z-values, which measure how many standard deviations a point is from the mean. Values above the mean have positive z-values; values below the mean have negative z-values.
Any normal distribution — with any mean and any standard deviation — can be converted ("standardized") into the standard normal distribution using the z-score transformation. This is what lets a single table (the z-table) answer probability questions for every possible normal distribution.
where x is any point on the original distribution, μ is the population mean, and σ is the population standard deviation.
Before doing any arithmetic, draw a quick bell curve, mark the mean in the center, mark your x-value(s) or z-value(s), and shade the area the question is asking for. This 10-second habit prevents almost every sign error and "which side of the table" mistake on this topic. If your shaded region is more than half the curve, your final probability must be greater than 0.50 — a fast sanity check.
The standard normal table in this course gives you P(0 ≤ Z ≤ z) — the area between the mean (z=0) and your z-value — not the cumulative area from negative infinity. You must add 0.50 to get the full P(Z < z) when z is positive.
The row gives z to the first decimal place; the column gives the second decimal place. Their intersection gives P(0 ≤ Z ≤ z). For example, row 2.0, column 0.00 gives 0.4772, so P(0 ≤ Z ≤ 2.00) = 0.4772.
Scenario: The gas mileage for a new SUV model is normally distributed with μ = 22 mpg, σ = 4. Find the probability an SUV gets less than 27 mpg, i.e., P(x < 27).
Step 1 — sketch and convert to z:
Step 2 — look up the table: P(0 ≤ Z ≤ 1.25) = 0.3944
Step 3 — add the left half (below the mean) since we want everything below z = 1.25:
Because the total area under the curve is 1 and the curve is symmetric: P(Z > z) = 1 − P(Z < z). This is the fastest way to handle "more than" or "greater than" questions once you already have P(Z < z) from the table.
Scenario: Using the same SUV gas mileage distribution (μ = 22, σ = 4), find the probability an SUV gets more than 27 mpg.
From the previous example, P(x < 27) = P(Z < 1.25) = 0.8944.
Scenario: Tips at a local restaurant are normally distributed with μ = $12, σ = $3. What is the probability a customer leaves a tip of less than $8?
Symmetry shortcut for negative z: P(Z ≤ −1.33) = P(Z ≥ 1.33) = 1 − P(Z ≤ 1.33). Looking up 1.33 in the table gives P(0 ≤ Z ≤ 1.33) = 0.4082.
So there is only about a 9.18% chance a customer tips less than $8.
A negative z means your x-value is below the mean. The table only lists positive z-values, but "probabilities associated with negative z-values equal the probabilities for the corresponding positive z-values" — meaning P(0 ≤ Z ≤ −1.33) uses the same table entry as P(0 ≤ Z ≤ 1.33) = 0.4082. The mistake is forgetting whether to add 0.50 or subtract from 0.50 — always sketch the curve, shade the actual region asked for, and match the shaded area's size (more or less than half?) with your arithmetic before trusting the number.
To find the probability between two z-values (or two x-values), find the table area for each z-value separately, then subtract:
Scenario: Same SUV gas mileage distribution (μ = 22, σ = 4). Find the probability of one SUV getting between 23 and 27 mpg.
From the table: P(0 ≤ Z ≤ 0.25) = 0.0987. P(0 ≤ Z ≤ 1.25) = 0.3944.
So about 29.57% of SUVs get between 23 and 27 mpg.
If the range straddles the mean (one z negative, one z positive), add the two table areas instead of subtracting, since each area is measured from the center outward in opposite directions: P(−a < Z < b) = P(0 ≤ Z ≤ a) + P(0 ≤ Z ≤ b). Always let your sketch tell you whether to add or subtract — if the shaded region straddles the centerline, you add; if both boundaries are on the same side, you subtract.
Instead of "find the probability," reverse problems ask "find the cutoff value." These are solved backward: look up the desired area/probability inside the table body to find the closest matching z-value, then un-standardize it back to x using the rearranged z formula:
Scenario: An entrance exam is normally distributed with μ = 800, σ = 50. If the school wants to admit only the top 10% of students, what should the cutoff score be?
Step 1 — sketch: the top 10% is the tail to the right of the cutoff. That means 40% lies between the mean and the cutoff (since the right half of the curve is 50%, and we want to leave 10% beyond the cutoff): area = 0.50 − 0.10 = 0.40.
Step 2 — search the table body for the value closest to 0.4000. The closest entry is 0.3997, which corresponds to z = 1.28.
Step 3 — un-standardize:
Applicants scoring 864 or higher will be admitted.
Scenario: A cell phone battery's time between charges is normally distributed with σ = 1.5 hours. A manufacturer wants no more than 5% of batteries to require charging in 13 hours or less. What must the mean time between charges be?
Step 1 — sketch: we want only 5% of the curve to the left of x = 13. That leaves 45% of area between the (unknown) mean and x = 13, since half the curve is 50% and we're carving 5% off the left tail.
Step 2 — find z for area 0.45: the closest z-table match is 0.45, giving z ≈ −1.645 (negative because x = 13 is below the mean we are solving for).
Step 3 — rearrange z = (x − μ)/σ to solve for μ:
The mean time between charges must be at least 15.47 hours to meet this requirement.
The z-table only works with z-values, never raw x-values. A very common exam error is looking up a raw score like "27 mpg" directly in the table. You must always run the z = (x − μ)/σ conversion first — every single normal-distribution table lookup routes through a z-value, no exceptions.
Certain z-values come up so often (especially once you reach confidence intervals in later chapters) that it pays to memorize them now. These are the z-values that cut off a specific total area in the two tails combined (two-tailed) — i.e., the value of z such that the middle area equals the stated confidence level.
| Confidence Level | Critical z-value |
|---|---|
| 80% | 1.28 |
| 90% | 1.645 |
| 95% | 1.96 |
| 99% | 2.575 |
Notice z = 1.28 is exactly the value used in the entrance-exam "top 10%" example above (a one-tailed 80% cutoff), and z = 1.645 is exactly the value used in the battery example (a one-tailed 90% cutoff / one-tailed 5% tail). These four critical values (1.28, 1.645, 1.96, 2.575) are the most-used z-values on exams for both one-tailed percentile problems and two-tailed normal-curve area problems — memorize this table cold.
1. In any normal distribution, which three measures of center are always equal?
2. The standard normal distribution always has:
3. The correct formula for converting a raw value x to a z-score is:
4. If the z-table gives P(0 ≤ Z ≤ 1.50) = 0.4332, what is P(Z > 1.50)?
5. P(Z < −2.00) equals:
6. A student is finding the score that cuts off the top 5% of a normally distributed exam. What should they search for in the z-table?
7. Which of the following is required before you can look up any raw score x in the standard normal table?
8. The 95% confidence-level critical z-value (two-tailed) is approximately:
Q1. Package weights are normally distributed with μ = 50 kg and σ = 5 kg. Find the probability a randomly chosen package weighs more than 58 kg.
Answer:
From the table, P(0 ≤ Z ≤ 1.60) = 0.4452.
Q2. Using the same package-weight distribution (μ = 50, σ = 5), find the probability a package weighs between 45 kg and 55 kg.
Answer:
Since one z is negative and one is positive, the range straddles the mean, so we add both table areas: P(0 ≤ Z ≤ 1.00) = 0.3413 for each side.
(This matches the Empirical Rule: about 68% of values fall within 1 standard deviation of the mean.)
Q3. Delivery times are normally distributed with μ = 30 minutes and σ = 4 minutes. The company wants to advertise a delivery time such that only 10% of deliveries take longer. What time should they advertise?
Answer: We want only 10% of area to the right of the cutoff, so 40% of area lies between the mean and the cutoff (0.50 − 0.10 = 0.40). Searching the table body for 0.40 gives z ≈ 1.28.
The company should advertise a delivery time of about 35 minutes.
Z-Score Transformation (standardizing a value):
Un-standardizing (reverse problem — solving for x):
Symmetry Shortcuts:
Between Two Same-Side Z-Values (both positive or both negative):
Between Two Opposite-Side Z-Values (straddling the mean):
Table Convention Used in This Course:
Empirical Rule (Approximate Areas — Continuous Analog):
Common Confidence-Level Critical Z-Values:
| Term | Definition |
|---|---|
| Normal distribution | A continuous, unimodal, symmetrical bell-shaped probability distribution where the mean, median, and mode are all equal and the total area under the curve equals 1. |
| Standard normal distribution | The specific normal distribution with mean μ = 0 and standard deviation σ = 1, scaled in z-values. |
| Z-score | A standardized value measuring how many standard deviations a point x is from the mean: z = (x − μ)/σ. |
| Area under the curve | The region under the normal curve between two points, which represents the probability of the random variable falling in that range; total area under any normal curve equals 1. |