6.1 The Standard Normal Distribution

The standard normal distribution is a powerful tool for analyzing data. It uses z-scores to measure how far data points are from the mean in terms of standard deviations. This lets us compare values across different datasets easily.

The Empirical Rule is a key feature, showing how data clusters around the mean. It tells us that 68% of data falls within one standard deviation, 95% within two, and 99.7% within three. This helps predict where most data will lie in a normal distribution.

The Standard Normal Distribution

Z-score calculation for data deviation

• Calculate z-score using formula $z = \frac{x - \mu}{\sigma}$
  • $x$ represents individual data point value
  • $\mu$ represents population mean
  • $\sigma$ represents population standard deviation
• Z-score indicates number of standard deviations a data point is from the mean
  • Positive z-score signifies data point above the mean (right side of distribution)
  • Negative z-score signifies data point below the mean (left side of distribution)
• Example calculation: Given $x = 85$, $\mu = 75$, and $\sigma = 5$, $z = \frac{85 - 75}{5} = 2$
  • Interpretation: The data point 85 is 2 standard deviations above the mean (75)

Empirical Rule for normal distributions

• Empirical Rule (68-95-99.7 Rule) applies to normally distributed data
  • 68% of data within 1 standard deviation of mean ($\mu \pm 1\sigma$)
  • 95% of data within 2 standard deviations of mean ($\mu \pm 2\sigma$)
  • 99.7% of data within 3 standard deviations of mean ($\mu \pm 3\sigma$)
• Calculate percentage of data within specific range using z-scores and Empirical Rule
  • Example: Percentage of data between 70 and 80 with $\mu = 75$ and $\sigma = 5$
    • Calculate z-scores: $z_{70} = \frac{70 - 75}{5} = -1$ and $z_{80} = \frac{80 - 75}{5} = 1$
    • Range spans -1 to 1 standard deviations from mean
    • Empirical Rule states 68% of data falls within this range (-1σ to 1σ)
• The area under the curve between two z-scores represents the probability of data falling within that range

Interpretation of positive vs negative z-scores

• Z-scores represent relative position of data point compared to mean
  • Positive z-score indicates data point above mean (right side)
    • Example: $z = 1.5$ means data point is 1.5 standard deviations above mean
  • Negative z-score indicates data point below mean (left side)
    • Example: $z = -2$ means data point is 2 standard deviations below mean
• Absolute value of z-score represents distance from mean in standard deviations
  • Larger absolute z-scores indicate data points further from mean
    • Example: $z = 2.5$ further from mean than $z = 1.2$ (2.5σ vs 1.2σ)
• Z-scores enable comparison of relative positions across different normal distributions
  • Example: $z = 1.5$ in distribution A is equivalent to $z = 1.5$ in distribution B

Additional Concepts in Normal Distribution

• The probability density function describes the shape of the normal distribution curve
• The central limit theorem states that the sampling distribution of the sample mean approaches a normal distribution as the sample size increases
• Standard error is the standard deviation of the sampling distribution of a statistic
• A normal probability plot can be used to assess whether a dataset follows a normal distribution