13.1 Measures of Central Tendency, Dispersion & Weighted Averages

Measures of Central Tendency, Dispersion & Weighted Averages

Quick Answer: On the WEST-B Mathematics subtest (Objective 0016), statistics questions evaluate your ability to compute and interpret summary statistics. Key formulas include: Mean ($\bar{x} = \frac{\sum x}{n}$), Median (middle value of sorted list; average of two middle values if $n$ is even), Mode (most frequent value), Range ($\text{Max} - \text{Min}$), Interquartile Range ($\text{IQR} = Q_3 - Q_1$), and Weighted Average ($\bar{x}_w = \frac{\sum (w_i \cdot x_i)}{\sum w_i}$). If data contains extreme outliers or skewness, the median is the most reliable measure of center because the mean is heavily distorted by extreme values.

---

1. Measures of Central Tendency

A measure of central tendency is a single summary statistic that identifies the center or typical value of a probability distribution or dataset.

+---------------------------------------------------------------------------------------------------+
|                             MEASURES OF CENTRAL TENDENCY SUMMARY                                  |
|                                                                                                   |
|   +-------------------+----------------------------+------------------------------------------+   |
|   | MEASURE           | MATHEMATICAL DEFINITION    | RESISTANCE TO OUTLIERS / BEST USE        |   |
|   +-------------------+----------------------------+------------------------------------------+   |
|   | Mean (x̄)          | Sum of values divided by   | NOT resistant; pulled by outliers.       |   |
|   |                   | count: Σx / n              | Best for symmetric, unskewed data.       |   |
|   +-------------------+----------------------------+------------------------------------------+   |
|   | Median            | Positional midpoint of an  | Highly RESISTANT (robust) to outliers.   |   |
|   |                   | ordered dataset            | Best for skewed data (salaries, housing).|   |
|   +-------------------+----------------------------+------------------------------------------+   |
|   | Mode              | Most frequently occurring  | Resistant; applicable to both numerical  |   |
|   |                   | value(s)                   | and categorical (nominal) data.          |   |
|   +-------------------+----------------------------+------------------------------------------+   |
+---------------------------------------------------------------------------------------------------+

A. The Arithmetic Mean

The mean (denoted $\bar{x}$ for a sample or $\mu$ for a population) is the arithmetic average: $\bar{x} = \frac{x_1 + x_2 + \dots + x_n}{n} = \frac{\sum_{i=1}^n x_i}{n}$

• Key Property: The sum of deviations from the mean is always zero: $\sum (x_i - \bar{x}) = 0$.
• Limitation: Every value in the dataset contributes equally to the sum, making the mean sensitive to extreme values (outliers).

B. The Median

The median is the value separating the higher half from the lower half of an ordered data sample.

1. Step 1: Sort the dataset in ascending numerical order.
2. Step 2: Determine the sample size $n$.
   • If $n$ is odd: The median is the single middle element at index position $\frac{n + 1}{2}$. $\text{Example: } \{3, 7, 9, 12, 18\} \implies n = 5 \implies \text{Position } 3 \implies \text{Median} = 9$
   • If $n$ is even: The median is the arithmetic average of the two middle elements at positions $\frac{n}{2}$ and $\frac{n}{2} + 1$. $\text{Example: } \{4, 8, 11, 15, 20, 26\} \implies n = 6 \implies \text{Average of 3rd and 4th} = \frac{11 + 15}{2} = 13$

C. The Mode

The mode is the data value that appears with the greatest frequency.

• Unimodal: Exactly one value occurs most frequently (e.g., ${2, 3, 5, 5, 8} \implies \text{Mode} = 5$).
• Bimodal: Two distinct values tie for the highest frequency (e.g., ${1, 4, 4, 7, 9, 9} \implies \text{Modes} = 4 \text{ and } 9$).
• Multimodal: Three or more values tie for highest frequency.
• No Mode: All values occur with equal frequency (e.g., ${10, 20, 30, 40, 50} \implies \text{No mode}$). Mode is the only measure of central tendency usable for categorical data (e.g., favorite color).

---

2. Distribution Shape & Outlier Effects on Center

When extreme values or asymmetric tails exist in a distribution, the mean, median, and mode separate in predictable patterns.

                               DISTRIBUTION SKEWNESS MATRIX

     Negative Skew (Left-Skewed)           Symmetric (Bell-Shaped)          Positive Skew (Right-Skewed)
            Long Left Tail                      Balanced Curve                    Long Right Tail
                 |                                    |                                    |
                 |      /\                            |     /\                             |  /\
                 |     /  \                           |    /  \                            | /  \
                 |    /    \                          |   /    \                           |/    \
             ____|___/______\_____                ____|__/______\__                    ____/______\_____
               Mean Median Mode                     Mean = Median = Mode                 Mode Median Mean
               (Mean < Median)                                                           (Mean > Median)

+---------------------------------------------------------------------------------------------------+
|                             SKEWNESS & MEASURE COMPARISON MATRIX                                  |
|                                                                                                   |
|   +-----------------------+-----------------------------+-------------------------------------+   |
|   | DISTRIBUTION SHAPE    | MATHEMATICAL RELATIONSHIP   | RECOMMENDED MEASURE OF CENTER       |   |
|   +-----------------------+-----------------------------+-------------------------------------+   |
|   | Symmetric             | Mean ≈ Median ≈ Mode        | Mean (captures total data magnitude)|   |
|   +-----------------------+-----------------------------+-------------------------------------+   |
|   | Right-Skewed          | Mode < Median < Mean        | Median (mean is pulled upward by    |   |
|   | (Positive Skew)       | (Mean > Median)             | high outliers)                      |   |
|   +-----------------------+-----------------------------+-------------------------------------+   |
|   | Left-Skewed           | Mean < Median < Mode        | Median (mean is pulled downward by  |   |
|   | (Negative Skew)       | (Mean < Median)             | low outliers)                       |   |
|   +-----------------------+-----------------------------+-------------------------------------+   |
+---------------------------------------------------------------------------------------------------+

Exam Strategy Tip: Whenever a WEST-B test question describes real-world contexts with extreme positive skewness—such as annual household incomes, CEO compensation, or home prices—the median is virtually always the most appropriate and representative measure of central tendency.

---

3. Measures of Dispersion (Spread)

Measures of dispersion quantify how spread out or clustered data values are relative to each other and the center.

+---------------------------------------------------------------------------------------------------+
|                                 MEASURES OF DISPERSION MATRIX                                     |
|                                                                                                   |
|   +-----------------------+-----------------------------+-------------------------------------+   |
|   | MEASURE               | FORMULA                     | KEY CONCEPT / PROPERTY              |   |
|   +-----------------------+-----------------------------+-------------------------------------+   |
|   | Range                 | Max - Min                   | Total span; extremely sensitive to  |   |
|   |                       |                             | outliers at both ends               |   |
|   +-----------------------+-----------------------------+-------------------------------------+   |
|   | Interquartile Range   | IQR = Q3 - Q1               | Spread of middle 50%; highly        |   |
|   | (IQR)                 |                             | resistant to extreme outliers       |   |
|   +-----------------------+-----------------------------+-------------------------------------+   |
|   | Variance (s² or σ²)   | s² = Σ(x - x̄)² / (n - 1)    | Average squared deviation from mean;|   |
|   |                       | σ² = Σ(x - μ)² / N          | measured in squared units           |   |
|   +-----------------------+-----------------------------+-------------------------------------+   |
|   | Standard Deviation    | s = √[Σ(x - x̄)² / (n - 1)]   | Average distance from mean; same    |   |
|   | (s or σ)              | σ = √[Σ(x - μ)² / N]        | units as raw data (σ = 0 if uniform)|   |
|   +-----------------------+-----------------------------+-------------------------------------+   |
+---------------------------------------------------------------------------------------------------+

Calculating Quartiles and IQR

1. Arrange data in ascending order and find the overall median ($Q_2$).
2. The First Quartile ($Q_1$) is the median of the lower half of the data (below $Q_2$).
3. The Third Quartile ($Q_3$) is the median of the upper half of the data (above $Q_2$).
4. $\text{IQR} = Q_3 - Q_1$.

Conceptual Understanding of Standard Deviation

• A dataset with tightly clustered values (e.g., ${49, 50, 50, 51}$) has a small standard deviation.
• A dataset with widely dispersed values (e.g., ${10, 30, 70, 90}$) has a large standard deviation.
• If all numbers in a dataset are identical (e.g., ${7, 7, 7, 7}$), the standard deviation is zero ($s = 0$).

---

4. Weighted Averages

A weighted average is used when different values in a dataset carry unequal importance, frequency, or proportional weight.

$\bar{x}_w = \frac{\sum_{i=1}^k (w_i \cdot x_i)}{\sum_{i=1}^k w_i} = \frac{w_1 x_1 + w_2 x_2 + \dots + w_k x_k}{w_1 + w_2 + \dots + w_k}$

If the weights are percentages that sum to $100%$ ($1.00$), the denominator equals $1$, simplifying to: $\bar{x}_w = \sum (w_i \cdot x_i)$

Standard Course Grade Weighting Breakdown

---

5. Missing Data Point Problems (Target Averages)

WEST-B exams regularly include word problems where an examinee must find the required score on a future test or project to achieve a target overall average.

General Solution Strategy

1. Identify the total number of items ($n$) including the missing item.
2. Calculate the required total sum: $\text{Target Sum} = n \times \text{Target Mean}$.
3. Calculate the current sum of known data points: $\text{Current Sum} = x_1 + x_2 + \dots + x_{n-1}$.
4. Subtract to find the missing value: $x_n = \text{Target Sum} - \text{Current Sum} = (n \cdot \bar{x}_{\text{target}}) - \sum x_{\text{known}}$

---

6. Step-by-Step Worked Statistical Problems

Problem 1: Step-by-Step Five-Number Summary & Spread Calculation

Problem: A classroom of 11 students scored the following on a mathematics quiz: $\{72, 85, 94, 68, 90, 88, 76, 82, 98, 88, 79\}$ Find the mean, median, mode, range, and interquartile range (IQR).

Step-by-Step Solution:

1. Sort data in ascending order: $68, 72, 76, 79, 82, 85, 88, 88, 90, 94, 98 \quad (n = 11)$
2. Calculate Mean: $\sum x = 68 + 72 + 76 + 79 + 82 + 85 + 88 + 88 + 90 + 94 + 98 = 920$ $\bar{x} = \frac{920}{11} \approx 83.64$
3. Find Median ($Q_2$): Since $n = 11$ (odd), the median is at position $\frac{11 + 1}{2} = 6^{\text{th}}$ position. $\text{Median } (Q_2) = 85$
4. Find Mode: The score $88$ appears twice; all other scores appear once. $\text{Mode} = 88$
5. Find Range: $\text{Range} = \text{Max} - \text{Min} = 98 - 68 = 30$
6. Find Quartiles and IQR:
   • Lower half (first 5 elements): ${68, 72, 76, 79, 82} \implies Q_1 = 76$ ($3^{\text{rd}}$ position).
   • Upper half (last 5 elements): ${88, 88, 90, 94, 98} \implies Q_3 = 90$ ($3^{\text{rd}}$ position of upper half). $\text{IQR} = Q_3 - Q_1 = 90 - 76 = 14$

---

Problem 2: Missing Final Exam Score for a Target Course Grade

Problem: Marcus has taken 4 history exams with scores of $82, 78, 90,$ and $86$. His final exam counts as two regular exam grades (equivalent to two identical test scores). What score must Marcus earn on the final exam to achieve an overall course average of exactly $85$ across all tests?

Step-by-Step Solution:

1. Determine total effective test count:
   • $4$ regular tests $+ 2$ test weights for the final $= 6$ total test components ($n = 6$).
2. Calculate required target total points: $\text{Target Sum} = 6 \times 85 = 510\text{ points}$
3. Calculate sum of the first 4 exams: $\text{Current Sum} = 82 + 78 + 90 + 86 = 336\text{ points}$
4. Set up equation for final exam score $F$: $336 + 2F = 510$ $2F = 510 - 336 = 174$ $F = \frac{174}{2} = 87$
5. Verify: $\frac{82 + 78 + 90 + 86 + 87 + 87}{6} = \frac{510}{6} = 85.0$ Marcus needs an $87$ on the final exam.

---

Problem 3: Weighted Mixture / Concentration Calculation

Problem: A science lab technician mixes $300\text{ mL}$ of a $20%$ saline solution, $500\text{ mL}$ of a $40%$ saline solution, and $200\text{ mL}$ of a $70%$ saline solution. What is the concentration percentage of the resulting $1,000\text{ mL}$ mixture?

Step-by-Step Solution:

1. Calculate pure saline contributed by each solution:
   • Solution 1: $300\text{ mL} \times 0.20 = 60\text{ mL pure saline}$
   • Solution 2: $500\text{ mL} \times 0.40 = 200\text{ mL pure saline}$
   • Solution 3: $200\text{ mL} \times 0.70 = 140\text{ mL pure saline}$
2. Sum total pure saline and total volume: $\text{Total Saline} = 60 + 200 + 140 = 400\text{ mL}$ $\text{Total Volume} = 300 + 500 + 200 = 1,000\text{ mL}$
3. Compute weighted concentration: $\text{Concentration} = \frac{400\text{ mL}}{1,000\text{ mL}} = 0.40 = 40\%$ The resulting mixture has a $40%$ concentration.