Distribution (Intuition) Math Example 1

Follow the full solution, then compare it with the other examples linked below.

Example 1

easy
Describe the distribution of heights of adult men (approximately normally distributed with mean 70 inches and SD 3 inches). What percentage of men are between 64 and 76 inches tall?

Solution

  1. 1
    Normal distribution: symmetric, bell-shaped, centered at ฮผ=70\mu = 70 inches
  2. 2
    64 inches is 64โˆ’703=โˆ’2\frac{64-70}{3} = -2 SD below the mean; 76 inches is 76โˆ’703=+2\frac{76-70}{3} = +2 SD above the mean
  3. 3
    Empirical rule: approximately 95% of data falls within ฮผยฑ2ฯƒ\mu \pm 2\sigma
  4. 4
    Therefore approximately 95% of men are between 64 and 76 inches tall

Answer

Approximately 95% of men have heights between 64 and 76 inches.
The normal distribution is described by its mean (center) and standard deviation (spread). The 68-95-99.7 rule: 68% within ยฑ1ฯƒ, 95% within ยฑ2ฯƒ, 99.7% within ยฑ3ฯƒ. This makes normal distributions highly predictable.

About Distribution (Intuition)

A distribution describes how data values are spread out across their range โ€” which values occur, how often, and whether the data is symmetric or skewed.

Learn more about Distribution (Intuition) โ†’

More Distribution (Intuition) Examples

Example 2 medium

Compare three distributions: (A) uniform (equal probability for all outcomes), (B) right-skewed (mos

Example 3 easy

A distribution has mean 50 and median 65. Is this distribution symmetric, left-skewed, or right-skew

Example 4 hard

The Central Limit Theorem says that sample means follow a normal distribution for large [formula], r

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