AP Statistics 6.1 Introducing Statistics: Why Be Normal? Study Notes

Introducing Statistics: Why Be Normal?

 Even when we draw multiple random samples from the same population, the shapes of those sample distributions often look different. Studying this variation helps us understand why the Normal distribution becomes central in statistics.

Questions Suggested by Variation in Shapes:

• Why do two random samples from the same population sometimes look very different?
• How much variation in the sample shape is expected due to chance?
• How does sample size affect the similarity of the sample distribution to the population distribution?
• Why does the Normal distribution appear so often when sampling?

Conceptual Explanation:

• Every sample is only a subset of the population. Random chance can make one sample look skewed, another look symmetric, and another look uniform, even if they all come from the same population.
• As the sample size increases, variation in shapes decreases — larger samples tend to better reflect the population distribution.
• The Central Limit Theorem explains that, regardless of the population’s shape, the sampling distribution of the sample mean becomes approximately Normal if the sample size is large enough.

Examples:

Example 1: Rolling Dice

• Population: outcomes from rolling a fair die (1 through 6, uniform distribution).
• Take Sample A (n = 10): results might cluster around 3 and 4 → histogram looks skewed right.
• Take Sample B (n = 10): results might include many 1’s and 2’s → histogram looks skewed left.
• Question Raised: Why do two samples from the same uniform population have such different shapes?

Example 2: Human Heights

• Population: all adult female heights in a country (approximately Normal, mean = 64 in, sd = 3 in).
• Take Sample A (n = 20): histogram might look somewhat symmetric, but with bumps or outliers.
• Take Sample B (n = 20): histogram might appear skewed because of a cluster of shorter or taller individuals.
• Question Raised: How large should the sample size be before the sample shape consistently resembles the population’s Normal shape?

Example 3: Coin Tossing

• Population: repeated coin flips (p = 0.5 heads, 0.5 tails).
• Sample A (n = 30 tosses): might show 18 heads, 12 tails → bar graph is slightly unbalanced.
• Sample B (n = 30 tosses): might show 15 heads, 15 tails → bar graph looks perfectly balanced.
• Question Raised: Why do two equally sized samples differ in how symmetric they appear?

Takeaway: Variation in sample shapes is normal. By asking these questions, we see the need for probability models like the Normal distribution to make sense of randomness and to develop reliable inference tools.