6.3 One-Variable Statistics: Center, Spread & Outliers
Key Takeaways
- The arithmetic mean ($\bar{x} = \frac{\sum x}{n}$) incorporates all values and is sensitive to extreme values, whereas the median (the middle value) is robust and resistant to outliers.
- Measures of spread quantify data variability: Range ($\text{Max} - \text{Min}$) and Standard Deviation ($\sigma$) are non-resistant to outliers, while the Interquartile Range ($\text{IQR} = Q_3 - Q_1$) is resistant.
- Distribution skewness determines the relationship between center metrics: symmetric distributions have $\text{Mean} \approx \text{Median}$, right-skewed distributions have $\text{Mean} > \text{Median}$, and left-skewed distributions have $\text{Mean} < \text{Median}$.
- In box-and-whisker plots, each of the four segments (left whisker, lower box, upper box, right whisker) contains exactly $25\%$ of the observations, regardless of the segment's visual length.
- Introducing an extreme high outlier increases the mean, range, and standard deviation substantially, while leaving the median and IQR virtually unchanged.
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.
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 $x$:
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.
| Quiz Score ($x$) | Number of Students ($f$) | Product ($f \cdot x$) |
|---|---|---|
| 1 | 2 | $2 \times 1 = 2$ |
| 2 | 3 | $3 \times 2 = 6$ |
| 3 | 7 | $7 \times 3 = 21$ |
| 4 | 5 | $5 \times 4 = 20$ |
| 5 | 3 | $3 \times 5 = 15$ |
| Total | $N = 20$ | $\sum (f \cdot x) = 64$ |
Calculate the Mean and Median:
- Mean ($\bar{x}$):
- 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, $\text{Median} = 3$.
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).
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:
- Identify the quartiles: $Q_1 = 68$ (25th percentile) and $Q_3 = 86$ (75th percentile).
- The range between $Q_1$ and $Q_3$ is the IQR, which contains exactly $50%$ of the data.
- Compute student count: $50% \text{ of } 80 = 0.50 \times 80 = 40\text{ students}$.
4. Distribution Shapes & The Mean-Median Relationship
The shape of a distribution dictates how its mean and median compare.
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| 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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| Distribution Shape | Mean vs. Median | Tail Orientation | Typical Context |
|---|---|---|---|
| Symmetric | $\text{Mean} \approx \text{Median}$ | Equal tails on both sides | Standardized test scores, heights |
| Right-Skewed (Positively Skewed) | $\text{Mean} > \text{Median}$ | Long tail extends to the right (high values) | Household incomes, home prices, CEO salaries |
| Left-Skewed (Negatively Skewed) | $\text{Mean} < \text{Median}$ | Long tail extends to the left (low values) | Easy exam scores, age at retirement |
5. Outlier Sensitivity: Resistant vs. Non-Resistant Statistics
An outlier is an observation that lies an abnormal distance from other values in the dataset.
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| 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 $$400{,}000$ and a median price of $$390{,}000$. A mansion priced at $$5{,}000{,}000$ is added to the listings. How do the mean and median respond?
Step-by-Step Solution:
- Original Mean Sum: $10 \times 400{,}000 = $4{,}000{,}000$.
- New Mean with 11th Home: The mean more than doubles (increases by over $$418{,}000$).
- 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.
- Conclusion: Mean is heavily distorted; median remains representative of the typical home price.
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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[!TIP] Expanding Frequency Tables in Desmos: If a problem gives a frequency table (e.g., score 3 with frequency 7, score 4 with frequency 5), you can quickly construct the list in Desmos:
L = [1, 1, 2, 2, 2, 3, 3, 3, 3, 3, 3, 3, 4, 4, 4, 4, 4, 5, 5, 5]Then typemean(L)andstats(L)to obtain the mean, median, IQR, and standard deviation in a single click.
A dataset containing the annual salaries of 120 employees at a startup company is strongly skewed to the right due to executive bonuses. Which statement must be true regarding the mean and median salaries of the employees?
A box-and-whisker plot represents test scores for 80 students. The minimum score is 45, the first quartile (Q1) is 68, the median (Q2) is 78, the third quartile (Q3) is 86, and the maximum score is 98. How many students scored between 68 and 86?
A dataset of 15 student quiz scores has a mean of 82, a median of 84, a range of 20, and a standard deviation of 5.8. A 16th student takes the make-up quiz and scores 20 (an extreme low outlier). Which of the following correctly describes how the summary statistics will change after including this score?