Standard Deviation, Empirical Rule, and Z-Scores Study Notes

Measures of Spread: Standard Deviation and Variance

• Distribution Shape and Central Tendency Selection:

  • Symmetric Distributions: Data distributions that are balanced around a central axis use the mean as the primary measure of central tendency.
  • Skewed Distributions: Data distributions that are asymmetric pull the mean away from the center, distorting the relationship between the mean and the median.

• Measures of Spread for Symmetric Data:

  • Standard Deviation: The primary metric used to quantify spread or dispersion for symmetric distributions.
  • Variance: A secondary metric for measuring spread, directly related to standard deviation.

• Notation and Symbols:

  • Sample Standard Deviation: Represented by the symbol $s$.
  • Sample Variance: Represented by the symbol $s^2$.
  • Population Standard Deviation: Represented by the lower-case Greek letter sigma ($σ$).
  • Population Variance: Represented by sigma squared ($σ^2$).

Standard Deviation Concepts and Manual Calculation Logic

• Definition of Deviation:

  • A deviation represents the directed distance between an individual data value ($x$) and the mean ($μ$ or ̅{x}):         $\text{Deviation} = x - \bar{x}$
  • Sign Conventions:
    • Positive Deviation: Indicates an observation is above the mean (e.g., a height of $6\,\text{ft}\,8\,\text{in}$ relative to a $5\,\text{ft}\,7\,\text{in}$ mean gives a positive deviation).
    • Negative Deviation: Indicates an observation is below the mean (e.g., a value $5\,\text{inches}$ below average yields a deviation of $-5\,\text{inches}$). A negative value does not imply a negative quality, only a positioning below average.

• Mathematical Rationale for the Standard Deviation Formula:

  • Sum of Deviations Principle: Simply averaging raw deviations fails because positive and negative distances perfectly cancel each other out, resulting in a sum of zero ($\sum (x - \bar{x}) = 0$).
  • Avoidance of Absolute Value: Converting negative deviations to positive using absolute values ($|x - \bar{x}|$) is mathematically disadvantageous because absolute value functions create non-differentiable points, making calculus applications difficult.
  • The Squaring Method: Squaring each deviation, $(x - \bar{x})^2$, eliminates negative signs while maintaining differentiability.
  • Process Steps:
    1. Compute the deviation for every individual observation ($x - \bar{x}$).
    2. Square each calculated deviation ($(x - \bar{x})^2$).
    3. Average the squared deviations (yielding the Variance).
    4. Take the principal square root of the variance to return to the original units of measurement (yielding the Standard Deviation).

• Conceptual and Visual Interpretation:

  • Standard deviation measures the rough average distance that observations fall from the mean.
  • Tightly Clustered Curve: Indicates low variability; features a small standard deviation.
  • Spread-Out (Pulled-Out) Curve: Indicates high variability; features a large standard deviation.
  • Utility of Spread: Standard deviation indicates how representative the center is of the entire dataset. When standard deviation is small, assuming all individuals equal the mean introduces minimal error. When standard deviation is large, assuming individuals equal the mean leads to significant inaccuracy.

Population vs. Sample Statistics in StatCrunch

• Default StatCrunch Computation:

  • Executing Stats $\rightarrow$ Summary Stats $\rightarrow$ Columns defaults strictly to sample statistics ($s$ and $s^2$).

• Population Parameter Computation Procedure:

  • To compute population parameters ($\u03C3$ and $σ^2$), unadjusted summary statistics must be explicitly selected.
  • Procedure:
    1. Navigate to Stats $\rightarrow$ Summary Stats $\rightarrow$ Columns.
    2. Select the target data column.
    3. In the selection box, scroll down, hold down the Ctrl key, and manually click Unadjusted Variance and Unadjusted Std. dev..
    4. Click Compute.
  • Application: Unadjusted metrics correspond to population values ($\u03C3^2$ and $σ$), whereas standard default output corresponds to sample values ($s^2$ and $s$).

The Empirical Rule (68-95-99.7 Rule) and Normal Curve Anatomy

• Curvature and Inflection Points:

  • Concave Down: The region near the peak of a normal distribution where the curve opens downward.
  • Concave Up: The outer tails of a normal distribution where the curve opens upward.
  • Inflection Point: The precise location on a normal curve where concavity transitions between concave down and concave up.
  • Geometric Identity: The horizontal distance from the mean to the inflection point is equal to exactly one standard deviation ($1\u03C3$).

• The Empirical Rule Breakdown:

  • Applicable only to bell-shaped, symmetric (normal) distributions.
  • $\pm 1\u03C3$ Range: Exactly $68\%$ of total observations lie within $1$ standard deviation of the mean ($\u03BC \pm 1\u03C3$).
  • $\pm 2\u03C3$ Range: Exactly $95\%$ of total observations lie within $2$ standard deviations of the mean ($\u03BC \pm 2\u03C3$).
  • $\pm 3\u03C3$ Range: Exactly $99.7\%$ of total observations lie within $3$ standard deviations of the mean ($\u03BC \pm 3\u03C3$).

• Interval Breakdown Percentages:

  • Between $\u03BC - 1\u03C3$ and $\u03BC$: $34\%$
  • Between $\u03BC$ and $\u03BC + 1\u03C3$: $34\%$
  • Between $\u03BC - 2\u03C3$ and $\u03BC - 1\u03C3$: $13.5\%$
  • Between $\u03BC + 1\u03C3$ and $\u03BC + 2\u03C3$: $13.5\%$
  • Between $\u03BC - 3\u03C3$ and $\u03BC - 2\u03C3$: $2.35\%$
  • Between $\u03BC + 2\u03C3$ and $\u03BC + 3\u03C3$: $2.35\%$
  • Below $\u03BC - 3\u03C3$ (lower tail): $0.15\%$
  • Above $\u03BC + 3\u03C3$ (upper tail): $0.15\%$

• Definition of Outlier:

  • An observation is formally classified as an outlier if it falls more than $2$ standard deviations away from the mean (beyond $\u03BC \pm 2\u03C3$).
  • Observations beyond $3$ standard deviations are extremely rare, occurring in approximately $3$ out of every $1,000$ cases ($0.3\%$).

• Example: American Men Weight Distribution:

  • Population Parameters: American men aged 25 to 70 with mean $\u03BC = 191\,\text{lbs}$ and standard deviation $\u03C3 = 26\,\text{lbs}$.
  • $\u03BC - 3\u03C3 = 113\,\text{lbs}$
  • $\u03BC - 2\u03C3 = 139\,\text{lbs}$
  • $\u03BC - 1\u03C3 = 165\,\text{lbs}$
  • $\u03BC = 191\,\text{lbs}$
  • $\u03BC + 1\u03C3 = 217\,\text{lbs}$
  • $\u03BC + 2\u03C3 = 243\,\text{lbs}$
  • $\u03BC + 3\u03C3 = 269\,\text{lbs}$
  • Inferences:
    • $68\%$ of men weigh between $165\,\text{lbs}$ and $217\,\text{lbs}$.
    • $95\%$ of men weigh between $139\,\text{lbs}$ and $243\,\text{lbs}$.
    • $99.7\%$ of men weigh between $113\,\text{lbs}$ and $269\,\text{lbs}$.

Applied Calculations Using the Empirical Rule

• Sample Distribution Parameters: Mean $\u03BC = 45$, Standard Deviation $\u03C3 = 3$

  • Interval Boundaries:
    • $36 = \u03BC - 3\u03C3$
    • $39 = \u03BC - 2\u03C3$
    • $42 = \u03BC - 1\u03C3$
    • $45 = \u03BC$
    • $48 = \u03BC + 1\u03C3$
    • $51 = \u03BC + 2\u03C3$
    • $54 = \u03BC + 3\u03C3$

• Problem 1: Percentage between $39$ and $54$:

  • Spans from $\u03BC - 2\u03C3$ to $\u03BC + 3\u03C3$.
  • Calculation: $95\%\,\text{(from 39 to 51)} + 2.35\%\,\text{(from 51 to 54)} = 97.35\%$.
  • Alternative Sum: $13.5\% + 34\% + 34\% + 13.5\% + 2.35\% = 97.35\%$.

• Problem 2: Percentage between $45$ and $51$:

  • Spans from $\u03BC$ to $\u03BC + 2\u03C3$.
  • Calculation: $34\% + 13.5\% = 47.5\%$.

• Problem 3: Percentage less than $42$:

  • Spans from $- \infty$ to $\u03BC - 1\u03C3$.
  • Calculation: $0.15\% + 2.35\% + 13.5\% = 16.0\%$.
  • Common Errors to Avoid:
    • Omitting the extreme tail percentage of $0.15\%$.
    • Misreading $0.15\%$ as $15\%$, leading to severe scale errors.

• Predicted vs. Actual Percentages:

  • Predicted Percentage: Computed via the Empirical Rule theoretical model.
  • Actual Percentage: Determined by sorting raw data values (e.g., using StatCrunch column sorting) and directly counting the empirical percentage of observations falling in the specified range. Sample values approximate, but rarely match, predicted values perfectly.

Standardized Values and Z-Scores

• Conceptual Overview:

  • The shape and relative area proportions of all normal distributions are identical regardless of unit measurements (e.g., body weights vs. IQ scores).
  • IQ Test Example: Mean $\u03BC = 100$, Standard Deviation $\u03C3 = 15$.
    • $\u03BC \pm 1\u03C3 = 85 \text{ to } 115$
    • $\u03BC \pm 2\u03C3 = 70 \text{ to } 130$
    • $\u03BC \pm 3\u03C3 = 55 \text{ to } 145$
  • Z-Score Function: Standardizes distributions to transform raw observations into universal metrics representing exact distance from the mean in standard deviation units.

• Z-Score Formula:     $z = \frac{x - \mu}{\sigma}$     Where:

  • $z$ = Standardized score
  • $x$ = Observed individual value
  • $\u03BC$ = Population mean
  • $\u03C3$ = Population standard deviation

• Z-Score Scale and Characteristics:

  • $z = 0$: Value is exactly equal to the mean.
  • $z = 1.0$: Value is exactly $1$ standard deviation above the mean (at the upper inflection point).
  • $z = -1.0$: Value is exactly $1$ standard deviation below the mean (at the lower inflection point).
  • Typical Bounds: Because $99.7\%$ of data resides between $z = -3$ and $z = 3$, standard z-scores almost always lie within the range $[-3, 3]$.
  • Computational Error Check: A calculated z-score like $10$ or $100$ indicates operational syntax error (e.g., failing to place the numerator in parentheses before dividing in a calculator).

• Practice Z-Score Calculations:

  • Geyser Eruption Time Example:
    • Given parameters: Mean duration $\u03BC = 245\,\text{seconds}$, observed value $x = 110\,\text{seconds}$.
    • Calculation:             $z = \frac{110 - 245}{\sigma} = -3.71$
    • Interpretation: An eruption duration of $110\,\text{seconds}$ has a z-score of $-3.71$, denoting an exceptionally short eruption that lies more than $3.7$ standard deviations below average (an extreme rarity, roughly $1$ in $10,000$).
  • Body Weight Evaluation Example:
    • Calculation yields $z = -0.30$
    • Interpretation: Represents a value $0.30$ standard deviations below average, which is completely typical and not a meaningful real-world deviation.