Normal Distribution Notes

Probability Density Curve

• A probability density curve describes the distribution of a variable.
• It indicates the proportion of the population within a given interval.
• The area under the curve between two values, $a$ and $b$, represents:
  • The proportion of the population with values between $a$ and $b$.
  • The probability that a randomly selected value from the population will be between $a$ and $b$.

Properties of Probability Density Curves

• The area above a single point is zero, meaning for a continuous random variable $x$, the probability that $x = a$ is 0 for any number $a$.
• P(a < x < b) = P(a \leq x \leq b), meaning including or excluding endpoints doesn't change the probability.
• The total area under the curve is 1, representing the entire population.

Normal Distribution

• Many statistical procedures rely on the normal curve.
• A population represented by a normal curve is normally distributed.
• The population mean determines the location of the peak of the normal curve.
• The population standard deviation measures the spread:
  • Large standard deviation: wide, flat curve.
  • Small standard deviation: tall, narrow curve.
• In a normal distribution:
  • Mean = Median = Mode
• Empirical Rule:
  • Approximately 68% of data within one standard deviation of the mean.
  • Approximately 95% of data within two standard deviations of the mean.
  • Approximately 99.7% of data within three standard deviations of the mean.

Standardization (Z-score)

• The z-score indicates the number of standard deviations a data value is above or below the mean.
• Standardization formula: $z = \frac{x - \mu}{\sigma}$, where:
  • $x$ is a value from a normal distribution.
  • $\mu$ is the mean of the distribution.
  • $\sigma$ is the standard deviation of the distribution.
• Example: For a woman with height $x = 67$ inches from a normal population with $\mu = 64$ inches and $\sigma = 3$ inches, the z-score is $z = \frac{67 - 64}{3} = 1$.
  • A value of 67 from N(64,3) is equivalent to a z-score of 1 from N(0,1).

Standard Normal Curve

• A normal distribution with a mean of 0 and a standard deviation of 1 is the standard normal distribution.
• Z-scores converted from a normal distribution follow a standard normal distribution.

Finding Area Between Z-scores in Excel

• Use NORM.S.DIST command to find area to the left of a z-score in a standard normal distribution.
• Syntax: NORM.S.DIST(z_score, cumulative)
  • z_score: The z-score.
  • cumulative: TRUE for cumulative probability.
• To find the area between two z-scores, subtract the smaller area from the larger area.
• For example, to find the area between z = -1.45 and z = 0.42:
  • Area to the left of -1.45: NORM.S.DIST(-1.45, TRUE) = 0.0735
  • Area to the left of 0.42: NORM.S.DIST(0.42, TRUE) = 0.6628
  • Area between -1.45 and 0.42: $0.6628 - 0.0735 = 0.5893$

Finding Z-score for a Given Area in Excel

• Use NORM.S.INV command to find the z-score corresponding to a given area to the left under the standard normal curve.
• Syntax: NORM.S.INV(probability)
  • probability: The area to the left of the desired z-score.
• Example: To find the z-score with an area of 0.26 to its left:
  • NORM.S.INV(0.26) = -0.6433