Normal Distribution

Normal Distribution

Introduction to Normal Distribution

This module covers the normal distribution, a fundamental concept in statistics, including how transformations affect its parameters, properties of its density curve, the empirical rule, z-scores, and the use of the standard normal table for probability calculations, as well as graphical assessment using Q-Q plots.

Shifting vs. Scaling

These are transformations applied to data that affect its statistical properties:

• Shifting (Center and Position): Adding or subtracting a constant value to each data point.

  • Effect on Mean: The mean changes by the amount of the constant added or subtracted.

  • Effect on Standard Deviation: The standard deviation (and thus spread) remains unchanged.

  • Example (Bonus Marks): If $X$ is the original mark and a bonus of $5$% (interpreted as $5$ percentage points) is given, making the new mark $S = X + 5$. The new mean score will be $\mu<em>S = \mu</em>X + 5$, while the standard deviation will remain the same, $\sigma<em>S = \sigma</em>X$.

• Scaling (Spread and Shape): Multiplying or dividing each data point by a constant value.

  • Effect on Mean: The mean is multiplied or divided by the same constant.

  • Effect on Standard Deviation: The standard deviation is also multiplied or divided by the absolute value of the same constant.

  • Effect on Variance: The variance is multiplied or divided by the square of the constant.

  • Formulas provided for standard deviation and variance when scaling by a factor $d$:

    • New standard deviation: $s<em>{new} = s</em>{original} \times d$

    • New variance: $s<em>{new}^2 = s</em>{original}^2 \times d^2$

  • Example (Scaling Marks): If $X$ is the original mark and each mark is scaled up by $5$%, meaning $S = 1.05 \times X$. The new mean score will be $\mu<em>S = 1.05 \times \mu</em>X$, and the new standard deviation will be $\sigma<em>S = 1.05 \times \sigma</em>X$. For example, if a student got $60$%, their new mark would be $60 \times 1.05 = 63$%.

Density Curve

A density curve is a graphical representation of the distribution of a continuous variable. It must satisfy specific conditions:

• Non-negativity: A density curve is always on or above the horizontal axis (i.e., its values are never negative, $f(x) \ge 0$).

• Total Area: The total area between the horizontal axis and under the density curve must equal $1$ (or $100$%), representing the total probability or proportion of observations.

• Probability as Area: The probability of an observation falling within an interval $(a, b)$, denoted as P(a < x < b), is equal to the area under the curve between $a$ and $b$. This also represents the percentage of observations that will fall in that interval.

• Continuous Variables Properties: For continuous variables, certain probability statements hold:

  • The probability of a single exact value is zero: $P(x = a) = 0$.

  • The inclusion or exclusion of endpoints does not change the probability for an interval: P(a < x < b) = P(a \le x \le b) = P(a < x \le b) = P(a \le x < b).

• Example (Density Curve from Page 5):

  • Verification of Area = 1 (by geometry): Assuming a uniform distribution from $0$ to $2$ with a height (density) of $0.5$. The area is $Base \times Height = (2 - 0) \times 0.5 = 2 \times 0.5 = 1$.

  • Probability that $X$ is less than $1$: P(X < 1) is the area from $0$ to $1$. $Area = (1 - 0) \times 0.5 = 1 \times 0.5 = 0.5$.

  • Probability that $X$ is greater than $1.5$: P(X > 1.5) is the area from $1.5$ to $2$. $Area = (2 - 1.5) \times 0.5 = 0.5 \times 0.5 = 0.25$.

Empirical Rule (68-95-99.7% Rule)

This rule applies specifically to data that follows a normal distribution, describing the approximate percentage of observations that fall within a certain number of standard deviations from the mean:

• Approximately $68$% of observations fall within $1$ standard deviation ($\pm 1\sigma$) of the mean ($\mu$).

• Approximately $95$% of observations fall within $2$ standard deviations ($\pm 2\sigma$) of the mean ($\mu$).

• Approximately $99.7$% of observations fall within $3$ standard deviations ($\pm 3\sigma$) of the mean ($\mu$).

• Example (Standardized Exam Completion Time):

  • Given: Mean ($\mu = 70$ minutes), Standard Deviation ($\sigma = 10$ minutes).

  • Percentage of students completing the exam in under an hour ($60$ minutes):

    • $60$ minutes is one standard deviation below the mean ($70 - 10 = 60$).

    • According to the empirical rule, $68$% of students complete between $60$ and $80$ minutes ($\mu \pm 1\sigma$).

    • The remaining $100\% - 68\% = 32\%$ are outside this range, split equally below $60$ and above $80$. So, $32\% / 2 = 16\%$ complete in under $60$ minutes.

  • Percentage of students completing the exam between $60$ and $70$ minutes:

    • This interval is from one standard deviation below the mean to the mean ($\mu - \sigma$ to $\mu$).

    • This represents exactly half of the $68$% interval: $68\% / 2 = 34\%$.

  • Time interval for the central $95$% of students:

    • The central $95$% of observations fall within $2$ standard deviations of the mean ($\mu \pm 2\sigma$).

    • The interval is $70 \pm (2 \times 10) = 70 \pm 20$.

    • Therefore, the central $95$% of students would be found between $50$ and $90$ minutes.

Standardized Score / z-score

The z-score is a measure of relative standing that indicates how many standard deviations an observation is away from the mean.

• Relative Standing: It positions a data point relative to the rest of the distribution.

• No Units: Z-scores are unitless, allowing comparison across different distributions.

• Interpretation: A z-score tells