normal-curve

Learning statistics is not about memorizing formulas — it’s about thinking with data.
Here are some strategies to make it easier.

1. Read Formulas in Two Ways

• Symbolic: $$\bar{X} = \frac{\Sigma X}{n}$$
• Words: “Mean = sum of scores / number of scores”

2. Practice by Hand First

• Work out a mean or variance with a small dataset.
• Then check with calculator/Excel.
• This builds intuition and confidence.

3. Draw Pictures

• Normal curve with shaded area
• Bar charts for group means
• Scatterplots for correlation
  Visuals make ideas stick.

4. Watch Out for Common Mistakes

• Mixing up SD and SEM
• Forgetting to subtract 1 for df
• Using a one-tailed test when two-tailed is needed

5. Use Short Sessions

• 10–15 minutes of practice each day beats one long cram.
• Try one formula or test per session.

6. Check Your Understanding

• Can you explain in words what the test does?
• Example: “t-test compares two means. ANOVA compares three or more.”

📱 QR: Online flashcards + short quiz (practice key terms & formulas)

Practice self-test quiz

In the space below, please find practice problems and self-test quizzes. For full access, please signup free.

The z-table gives areas (probabilities) under the standard normal curve (mean $$\mu=0$$, SD $$\sigma=1$$).
Use it after you standardize a score:

Standardization (z-score):
$$z=\frac{x-\mu}{\sigma}$$
In words: $$z=\frac{\text{score} - \text{mean}}{\text{standard deviation}}$$

What the z-table shows

Most tables list the area to the left of a z value (cumulative probability).

• Left area at $$z=0$$ is 0.5000 (half the curve).
• Far left (negative big z) approaches 0; far right (positive big z) approaches 1.

Quick recipes

1) Probability below a score (left tail)
Example: $$z=1.00$$ → table gives 0.8413.
Interpretation: $$P(Z \le 1.00)=0.8413$$ (84.13% below).

2) Probability above a score (right tail)
Use complement: $$P(Z \ge z)=1-\text{left area}$$.
Example: $$z=1.00 \Rightarrow P(Z \ge 1.00)=1-0.8413=0.1587.$$

3) Probability between two scores
Subtract left areas.
Example: between $$z= -0.50$$ (left area 0.3085) and $$z=1.20$$ (0.8849):
$$P(-0.50 \le Z \le 1.20)=0.8849-0.3085=0.5764.$$

4) From a raw score to probability
Test scores: $$\mu=100, \ \sigma=15$$. What % are below 115?
Standardize: $$z=\frac{115-100}{15}=1.00 \Rightarrow 0.8413 \ (\text{84.13%}).$$

5) From probability to raw score (percentile)
What score is the 90th percentile?
Find z with left area ≈ 0.9000 → $$z \approx 1.2816$$.
Convert back: $$x=\mu+z\sigma=100+(1.2816)(15)=119.22.$$

Tips

• For negative z, use the table’s symmetry: left area at $$-z$$ equals 1 − left area at $$+z$$.
• Rounding: two decimals is common (e.g., 1.23).
• Modern tools (calculator/Sheets/Python) can give exact p-values directly.

Visuals

Figure D.1 — Normal curve with area left of z = 1.00 shaded (0.8413).
Figure D.2 — Two-z shaded band for “between” probability.

📱 QR: Online z-calculator (type z or x, get areas instantly)

Practice self-test quiz

In the space below, please find practice problems and self-test quizzes. For full access, please signup free.

The normal curve (bell curve) is one of the most important concepts in statistics.
It is elegant, symmetrical, and central to probability and inference.
It appears whenever many small, independent factors combine: height, exam scores, measurement errors.

Properties of the Normal Curve

1. Symmetrical around the mean
2. One peak (unimodal)
3. Mean = Median = Mode
4. Total area under the curve = 1 (100%)

Formula for the Normal Distribution

Symbolic formula:
$$f(x) = \frac{1}{\sigma \sqrt{2\pi}} e^{-\frac{(x - \mu)^2}{2\sigma^2}}$$

Formula in words:
$$\text{Probability density} = \frac{1}{\text{standard deviation} \times \sqrt{2\pi}} \times e^{-\frac{(\text{score} - \text{mean})^2}{2 \times (\text{standard deviation})^2}}$$

Where:

• $$\mu$$ = mean
• $$\sigma$$ = standard deviation
• $$x$$ = a score

Standardization (z-scores)

Symbolic formula:
$$z = \frac{x - \mu}{\sigma}$$

Formula in words:
$$z = \frac{\text{score} - \text{mean}}{\text{standard deviation}}$$

A z-score tells us how many standard deviations a score is above or below the mean.

Key Percentages

Under the normal curve:

• About 68% of scores are within 1 standard deviation of the mean
• About 95% are within 2 standard deviations
• About 99.7% are within 3 standard deviations

This is called the 68–95–99.7 rule.

Drama Box — “The Goddess Normal Curve”

Imagine a temple where a perfect curve stands tall — balanced and symmetrical.

• At the center is the mean, the balance point.
• Half of the people (data) stand on each side.
• As you move further away, fewer remain.
• The Goddess teaches fairness: most scores are near the center, extreme scores are rare.

This image helps students remember the normal curve not as a dry formula, but as a principle of balance and probability.

Visuals

Figure L2.1 — The Normal Curve. Bell-shaped curve centered at the mean (μ).

Figure L2.2 — The 68–95–99.7 Rule. Normal curve with shaded regions ±1σ, ±2σ, ±3σ.

Figure L2.3 — z-score Example. Normal curve with shaded area to the left of z = 1.0, labeled 0.8413.

Why This Matters

The normal curve is the foundation of inferential statistics.

• It allows us to calculate probabilities.
• It underlies t-tests, ANOVAs, and confidence intervals.
• It lets us compare scores across different tests and scales.

Practice self-test quiz

In the space below, please find practice problems and self-test quizzes. For full access, please signup free.