6.3 One-Variable Statistics: Center, Spread & Outliers

6.3 One-Variable Statistics: Center, Spread & Outliers

One-variable statistics questions evaluate your ability to summarize, visualize, and compare single-variable datasets. On the Digital SAT, you will rarely need to perform tedious manual standard deviation calculations; instead, the exam tests your conceptual understanding of how center, spread, distribution shape, and outliers interact.

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1. Measures of Center: Mean vs. Median

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|                         MEASURES OF CENTRAL TENDENCY                        |
|                                                                             |
|   [ ARITHMETIC MEAN (x̄) ]            [ MEDIAN (M) ]                         |
|   - Formula: x̄ = (∑ x) / n           - The physical middle value (50th %ile)|
|   - Weighted: x̄ = ∑(f_i * x_i) / N   - Resistant / Robust to outliers       |
|   - Non-resistant (pulled by tails)  - Pos: (n + 1)/2 for odd n             |
|                                      - Pos: Avg of n/2 & (n/2 + 1) for even |
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Calculating Mean and Median from Frequency Tables

When data is presented in a frequency table, account for the frequency $f$ of each value $x$:

$$\bar{x} = \frac{\sum (f_i \cdot x_i)}{\sum f_i}$$

Worked Example 1: Frequency Table Calculations

Problem: The table below shows the distribution of scores on a 5-question math quiz taken by 20 students.

Calculate the Mean and Median:

1. Mean ($\bar{x}$): $$\bar{x} = \frac{64}{20} = 3.2$$
2. Median: For $n = 20$ students, the median is the average of the 10th and 11th values in ordered sequence:
   • Values 1 to 2: Score = 1
   • Values 3 to 5: Score = 2
   • Values 6 to 12: Score = 3 (both the 10th and 11th students fall in this group)
   • Thus, $\text{Median} = 3$.

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2. Measures of Spread: Range, IQR & Standard Deviation

Measures of spread quantify the degree of dispersion or variability among data points.

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|                          MEASURES OF DATA DISPERSION                        |
|                                                                             |
|   1. RANGE: Max - Min                                                       |
|      - Simplest measure of total spread. Highly sensitive to outliers.      |
|                                                                             |
|   2. INTERQUARTILE RANGE (IQR): Q3 - Q1                                     |
|      - Spread of the middle 50% of data. Resistant to outliers.             |
|                                                                             |
|   3. STANDARD DEVIATION (σ):                                                |
|      - Typical/average distance of data points from the arithmetic mean.    |
|      - Data clustered near the mean  --> Smaller standard deviation         |
|      - Data spread far from the mean --> Larger standard deviation          |
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Conceptual Comparison of Standard Deviation

You do not need the formula $\sigma = \sqrt{\frac{\sum (x - \bar{x})^2}{n}}$ on test day. Instead, compare datasets visually:

• Dataset A: $\{10, 10, 10, 10, 10\} \implies \sigma = 0$ (all values identical to mean).
• Dataset B: $\{8, 9, 10, 11, 12\} \implies \text{Small } \sigma$ (values tightly packed around 10).
• Dataset C: $\{1, 1, 10, 19, 19\} \implies \text{Large } \sigma$ (values concentrated at extreme boundaries).

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3. Visual Data Displays: Box Plots, Histograms & Dot Plots

Anatomy of a Box-and-Whisker Plot

A box plot displays the Five-Number Summary: Minimum, First Quartile ($Q_1$), Median ($Q_2$), Third Quartile ($Q_3$), and Maximum.

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|                         BOX-AND-WHISKER PLOT STRUCTURE                      |
|                                                                             |
|              |-------[=====|=====]-------|                                  |
|             Min     Q1    Median Q3     Max                                 |
|                                                                             |
|   |<-- 25% -->|<-- 25% -->|<-- 25% -->|<-- 25% -->|                         |
|   |           |<------ IQR (50%) ----->|          |                         |
|   |<------------------- Range (100%) ------------>|                         |
|                                                                             |
|   KEY RULE: Every section contains EXACTLY 25% of the total observations,   |
|             even if the physical length of the segment is longer or shorter.|
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Worked Example 2: Interpreting Box Plots

Problem: A box plot summarizes test scores for $80$ students: $\text{Min} = 45$, $Q_1 = 68$, $\text{Median} = 78$, $Q_3 = 86$, $\text{Max} = 98$. How many students scored between $68$ and $86$?

Step-by-Step Solution:

1. Identify the quartiles: $Q_1 = 68$ (25th percentile) and $Q_3 = 86$ (75th percentile).
2. The range between $Q_1$ and $Q_3$ is the IQR, which contains exactly $50\%$ of the data.
3. Compute student count: $50\% \text{ of } 80 = 0.50 \times 80 = 40\text{ students}$.

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4. Distribution Shapes & The Mean-Median Relationship

The shape of a distribution dictates how its mean and median compare.

+-----------------------------------------------------------------------------+
|                      DISTRIBUTION MORPHOLOGY & SKEWNESS                     |
|                                                                             |
|   [ SYMMETRIC / BELL ]         [ RIGHT-SKEWED / POSITIVE ]   [ LEFT-SKEWED ]|
|           /\                       /\                                  /\   |
|          /  \                     /  \                                /  \  |
|        /      \                  /    \________             ________/    \  |
|       /________\                /______________\           /______________\ |
|                                                                             |
|     Mean ≈ Median                Mean > Median              Mean < Median   |
|   (Balanced tails)          (Tail pulls Mean Right)    (Tail pulls Mean Left|
+-----------------------------------------------------------------------------+

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5. Outlier Sensitivity: Resistant vs. Non-Resistant Statistics

An outlier is an observation that lies an abnormal distance from other values in the dataset.

+-----------------------------------------------------------------------------+
|                       IMPACT OF OUTLIERS ON STATISTICS                      |
|                                                                             |
|   Statistic Type         Resistant to Outliers?      Behavior with Outlier  |
|   --------------------   -------------------------   ---------------------  |
|   Mean                   NO  (Non-Resistant)         Shifts toward outlier  |
|   Standard Deviation     NO  (Non-Resistant)         Increases significantly|
|   Range                  NO  (Non-Resistant)         Increases directly     |
|   Median                 YES (Resistant)             Minimal / No change    |
|   IQR (Q3 - Q1)          YES (Resistant)             Minimal / No change    |
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Worked Example 3: Adding an Extreme Outlier

Problem: A real estate firm lists 10 homes with an average price of $\text{\textdollar}400{,}000$ and a median price of $\text{\textdollar}390{,}000$. A mansion priced at $\text{\textdollar}5{,}000{,}000$ is added to the listings. How do the mean and median respond?

Step-by-Step Solution:

1. Original Mean Sum: $10 \times 400{,}000 = \text{\textdollar}4{,}000{,}000$.
2. New Mean with 11th Home: $$\text{New Mean} = \frac{4{,}000{,}000 + 5{,}000{,}000}{11} = \frac{9{,}000{,}000}{11} \approx \$818{,}182$$ The mean more than doubles (increases by over $\text{\textdollar}418{,}000$).
3. New Median: The median shifts only to the adjacent middle value in the ordered list (e.g., from the 5th/6th average to the 6th value), changing by only a few thousand dollars.
4. Conclusion: Mean is heavily distorted; median remains representative of the typical home price.

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6. Desmos List Statistics & Five-Number Summary Playbook

The built-in Desmos calculator features full support for list operations and descriptive statistics.

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|                         DESMOS LIST STATISTICS SYNTAX                       |
|                                                                             |
|   1. Define a List:                                                         |
|      L = [12, 15, 18, 22, 25, 30, 35, 42]                                   |
|                                                                             |
|   2. Compute Center & Spread Instantly:                                     |
|      mean(L)        --> Arithmetic mean (x̄)                                 |
|      median(L)      --> Middle value                                        |
|      stdev(L)       --> Sample standard deviation                           |
|      stdevp(L)      --> Population standard deviation                       |
|      mad(L)         --> Mean absolute deviation                             |
|                                                                             |
|   3. Instant Five-Number Summary & Box Plot:                                |
|      stats(L)       --> Returns [min, Q1, median, Q3, max]                  |
|      boxplot(L)     --> Renders visual interactive box plot on canvas       |
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