Comprehensive Study Notes on Descriptive Statistics, Density Curves, Normal Distributions, and Z-Scores

Recap of Descriptive Measures: Center and Spread

• Measures of Center:

  • Mode: The most frequently occurring value in a dataset.

  • Median: The exact middle value of an ordered distribution when values are sorted from smallest to largest ($50\text{th}$ percentile).

  • Mean: The arithmetic average of all data points in the distribution.

• Measures of Spread: Resistant vs. Non-Resistant Categories:

  • Resistant Spread Measures (Robust to extreme outliers and extreme peripheral values):

    • Five-Number Summary: Minimum, First Quartile ($Q_1$), Median ($Q_2$), Third Quartile ($Q_3$), and Maximum.

    • Interquartile Range ($IQR$): The distance between the $25\text{th}$ percentile ($Q_1$) and the $75\text{th}$ percentile ($Q_3$). Adding an extreme outlier at the fringe of the distribution does not substantially alter the $IQR$.

  • Non-Resistant Spread Measures (Highly sensitive to outliers):

    • Standard Deviation ($s$ or $\sigma$) and Variance ($s^2$ or $\sigma^2$).

    • Calculated by taking the deviation of every single value from the mean and squaring it. Squaring deviations weights larger peripheral distances disproportionately.

Activity Inequality and Public Health Case Study

• Study Design & Dataset:

  • Analyzed anonymized smartphone step-count data from participants across $111\,\text{countries}$ to evaluate worldwide physical activity patterns.

  • Research Problem: Determining why obesity rates vary significantly across nations despite similar national mean daily step counts.

• Findings on Mean vs. Spread:

  • The mean daily step count is not an effective predictor of national obesity rates.

  • Obesity rates are driven by activity inequality (the spread/distribution of physical activity across a population) rather than the center of the distribution.

  • Obesity concentrates primarily among the extreme lower tail of the physical activity distribution (highly inactive populations taking minimal steps).

• Statistical Explanatory Power ($R^2$):

  • $R^2$ Definition: A statistical metric representing the proportion of variance in an outcome variable predictable from an independent variable.

  • Activity Inequality $R^2$: $0.64$ (approximately $64\%$ of variance in national obesity rates is explained by activity inequality).

  • Mean Daily Steps $R^2$: $0.47$ (only $47\%$ of variance in national obesity rates is explained by mean daily steps).

• Comparing Two Hypothetical Countries:

  • Country A: High clustering around the mean step count ($6{,}000\,\text{steps}$). Low standard deviation, tight distribution, very small inactive tail, resulting in low obesity rates.

  • Country B: Identical mean step count ($6{,}000\,\text{steps}$), but high standard deviation and spread. Features a large population segment in the highly inactive tail alongside a highly active segment, resulting in higher obesity rates.

• Demographic Stratification & Sampling Biases:

  • Gender Stratification: While overall population distribution is approximately $50\%$ male and $50\%$ female, activity data is often unstratified by gender. Failing to break down data by gender hides gender gaps, risking public health interventions that amplify inequality.

  • Smartphone Sampling Biases:

    • Individual-Level Bias: Smartphone ownership favors younger and wealthier demographics.

    • Population Demographics: Older populations walk less on average due to age patterns. A healthy country with an aging population (e.g., Japan) may exhibit lower mean steps due to demographic composition while maintaining low obesity rates.

• Core Methodological Principles:

  1. Always report measures of spread alongside the mean.

  2. Extreme values (tails), rather than central values, often drive real-world systemic outcomes.

  3. Stratify descriptive statistics by sub-groups (e.g., gender, age, region) to uncover hidden patterns of inequality.

  4. Evaluate sample characteristics against overall population demographics to spot sampling bias.

Building Blocks of Spread: Deviation to Standard Deviation

• 1. Deviation:

  • Formula: $x - \bar{x}$ (sample) or $x - \mu$ (population).

  • Represents the distance and direction of a single data point from the mean. Can be positive or negative.

• 2. Squared Deviation:

  • Formula: $(x - \bar{x})^2$

  • Squares the distance of an observation from the mean. Ensures all values are positive and rescales units into squared terms.

• 3. Variance:

  • Formula (Population): $\sigma^2 = \frac{\sum (x - \mu)^2}{N}$

  • Formula (Sample): $s^2 = \frac{\sum (x - \bar{x})^2}{n - 1}$

  • The average squared deviation. Measured in squared units (e.g., $\text{hours}^2$), making direct intuitive interpretation difficult.

• 4. Standard Deviation:

  • Formula: $\sigma = \sqrt{\sigma^2}$ or $s = \sqrt{s^2}$

  • The square root of the variance. Returns the metric to the original measurement units, representing the typical distance of data points from the mean.

Linear Transformations of Data

• Core Rule: Linear transformations change the center and/or spread of a distribution, but NEVER change the underlying shape of the distribution.

• Types of Linear Transformations:

  • Shift Transformation (Adding/Subtracting a Constant $a$):

    • Formula: $x' = x + a$

    • Shifts measures of center (mean, median, mode) by $a$.

    • Measures of spread (standard deviation, $IQR$, variance) remain unchanged.

    • Example: Converting temperature offsets (e.g., shifting Fahrenheit to Celsius scales).

  • Rescaling Transformation (Multiplying/Dividing by a Constant $b$):

    • Formula: $x' = b \times x$

    • Rescales measures of center by $b$ and measures of spread (standard deviation, $IQR$) by $|b|$.

    • Example: Converting miles to kilometers, cents to Canadian dollars, or rescaling an exam score out of $1{,}200\,\text{points}$.

  • Combined Transformation:

    • Formula: $x' = a + b \times x$

    • Rescales spread by $|b|$ and shifts center by $a$.

Density Curves and Normal Distributions

• Density Curve Definition: A smooth mathematical curve that models and approximates the shape of an empirical data histogram (e.g., standard test scores of $947\,\text{seventh-graders}$ on the Iowa Test of Basic Skills).

• Two Mandatory Rules for Density Curves:

  1. The curve must lie entirely on or above the horizontal axis ($f(x) \ge 0$ for all $x$). Negative concentration of observations is impossible.

  2. The total area underneath the density curve must equal exactly $1$ ($100\%$ of all observations).

• Mean and Median on Density Curves:

  • Symmetric Unimodal Curve: Mean = Median = Mode, located at the central peak.

  • Right-Skewed Curve: Tail extends right; the mean is pulled rightward by extreme values ($\text{Mean} > \text{Median}$).

  • Left-Skewed Curve: Tail extends left; the mean is pulled leftward by extreme values ($\text{Mean} < \text{Median}$).

• Normal Distribution Parameters:

  • Notation: $N(\mu, \sigma)$, where $\mu$ is the mean and $\sigma$ is the standard deviation.

  • Determined entirely by $\mu$ (location of peak/center) and $\sigma$ (width/spread).

  • Modifying $\mu$ shifts the curve horizontally without altering shape.

  • Modifying $\sigma$ alters spread: smaller $\sigma$ produces a tall, narrow peak; larger $\sigma$ produces a broad, flat curve.

The 68-95-99.7 Empirical Rule

• Rule Definition: For any normally distributed variable $N(\mu, \sigma)$:

  • Approximately $68\%$ of observations lie within $1\sigma$ of the mean: $[\mu - \sigma, \mu + \sigma]$.

  • Approximately $95\%$ of observations lie within $2\sigma$ of the mean: $[\mu - 2\sigma, \mu + 2\sigma]$.

  • Approximately $99.7\%$ of observations lie within $3\sigma$ of the mean: $[\mu - 3\sigma, \mu + 3\sigma]$.

• Example 1: Women's Heights ($N(64.5, 2.5)$):

  • $\mu = 64.5\,\text{inches}$, $\sigma = 2.5\,\text{inches}$.

  • $68\%$ Interval: $64.5 \pm 2.5 \implies [62.0\,\text{inches}, 67.0\,\text{inches}]$.

  • $95\%$ Interval: $64.5 \pm 2(2.5) \implies [59.5\,\text{inches}, 69.5\,\text{inches}]$.

    • The outer $5\%$ is split into $2.5\%$ below $59.5\,\text{inches}$ and $2.5\%$ above $69.5\,\text{inches}$.

  • $99.7\%$ Interval: 64.5 \pm 3(2.5) \n\implies [57.0\,\text{inches}, 72.0\,\text{inches}].

• Example 2: NAEP Eighth-Grade Mathematics Test ($N(282, 40)$):

  • $\mu = 282\,\text{points}$, $\sigma = 40\,\text{points}$.

  • $95\%$ Range ($2\sigma$): $282 \pm 2(40) \implies [202\,\text{points}, 362\,\text{points}]$.

  • $99.7\%$ Range ($3\sigma$): $282 \pm 3(40) \implies [162\,\text{points}, 402\,\text{points}]$.

  • Evaluating Score of $322\,\text{points}$:

    • Sitting at $282 + 40 = 322\,\text{points}$ puts the score at exactly $+1\sigma$ above the mean.

    • Proportion of students scoring below $322\,\text{points}$ = $50\% + 34\% = 84\%$.

Standardizing Data and Z-Scores

• Definition: Standardizing converts observations from their original units into standardized units of standard deviations away from the mean.

• Standard Normal Distribution:

  • Notation: $N(0, 1)$, where $\mu = 0$ and $\sigma = 1$

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

• Interpretation of Z-Scores:

  • $z = 0$: Observation equals the mean.

  • $z > 0$: Observation lies above the mean.

  • $z < 0$: Observation lies below the mean.

  • $|z| \ge 2.0$ or $|z| \ge 3.0$: Indicates an unusual or rare observation.

• Step-by-Step Z-Score Calculation (Women's Heights $N(64.5, 2.5)$ ):

  • Step 1: Recenter (Subtract mean $\mu$).

  • Step 2: Rescale (Divide by standard deviation $\sigma$).

  • Case 1: Height $x = 68\,\text{inches}$:         $z = \frac{68 - 64.5}{2.5} = \frac{3.5}{2.5} = 1.40$         (Height is $1.40\,\text{standard deviations}$ above the mean).

  • Case 2: Height $x = 60\,\text{inches}$ ($5\,\text{feet}$):         $z = \frac{60 - 64.5}{2.5} = \frac{-4.5}{2.5} = -1.80$         (Height is $1.80\,\text{standard deviations}$ below the mean).

  • Case 3: Height $x = 66\,\text{inches}$:         $z = \frac{66 - 64.5}{2.5} = \frac{1.5}{2.5} = 0.60$         (Height is $0.60\,\text{standard deviations}$ above the mean).

Cumulative Proportions and Utilizing Table A

• Table A Definition: A standardized table listing cumulative proportions $P(Z < z)$, representing the area under the standard normal curve to the LEFT of a specific z-score.

• Upper Tail Rule: To calculate the proportion greater than a given z-score (area to the right), subtract the Table A cumulative value from 1:     $P(Z > z) = 1 - P(Z < z)$

• How to Read Table A:

  • The left column gives the z-score to the tenths place (e.g., $1.4$).

  • The top row gives the hundredths place (e.g., $0.07$).

  • The intersection provides the cumulative left-tail area.

  • Example: For $z = 1.47$, locate row $1.4$ and column $0.07$, yielding cumulative area $0.9292$ ($92.92\%$ of observations lie to the left).

• Worked Example: SAT Scores & NCAA Division I Eligibility:

  • Distribution: SAT scores for $1{,}400{,}000\,\text{students}$ modeled as $N(1026, 209)$.

  • NCAA Eligibility Threshold: Minimum score of $x = 820$.

  • Step 1: Compute Z-Score:         $z = \frac{820 - 1026}{209} = \frac{-206}{209} \approx -0.99$

  • Step 2: Find Cumulative Area in Table A:         Lookup for $z = -0.99$ gives $P(Z < -0.99) = 0.1611$ ($16.11\%$ score below $820$).

  • Step 3: Calculate Qualifying Proportion ($\ge 820$):         $P(Z \ge -0.99) = 1 - 0.1611 = 0.8389 \implies 83.89\%$         Approximately $84\%$ of test-takers qualify.

  • Step 4: Range Calculation ($720\,\text{to}\,820\,\text{points}$):         Subtract cumulative area at $x = 720$ from cumulative area at $x = 820$, yielding approximately $9\%$ of test-takers in this range.

Inverse Lookup Problems: Determining Threshold Scores

• Concept: Finding an unknown raw value $x$ given a target percentile or percentage.

• Problem: Determine the minimum SAT score required to reach the top $10\%$ of all test takers ($90\text{th}$ percentile, cumulative area $0.9000$).

• Step 1: Locate Target Cumulative Area in Table A:

  • Search the interior of Table A for the value closest to $0.9000$

  • $z = 1.28$ gives cumulative area $0.8997$

  • $z = 1.29$ gives cumulative area $0.9015$

  • Select $z = 1.28$ as the closest approximation to $0.9000$

• Step 2: Un-Standardize to Solve for Raw Score $x$:     $x = \mu + z \times \sigma$     $x = 1026 + (1.28 \times 209) = 1026 + 267.52 = 1293.52$     A student must score approximately $1294\,\text{points}$ to place in the top $10\%$.

Summary Quick Reference

• Deviation: $x - \mu$ (Distance from mean; positive or negative).

• Squared Deviation: $(x - \mu)^2$ (Eliminates negative signs; weights extreme outliers heavily).

• Variance: $\sigma^2 = \frac{\sum (x - \mu)^2}{N}$ (Average squared deviation; measured in squared units).

• Standard Deviation: $\sigma = \sqrt{\sigma^2}$ (Typical distance from mean; in original units).

• Z-Score: $z = \frac{x - \mu}{\sigma}$ (Standardized distance from mean in standard deviation units).