# Study Notes: Measures of Central Tendency and Spread ## The Center: Weighted Mean When some values in a dataset are more important than others, we calculate the **weighted mean**. The formula is: $$ \mu = \frac{\sum w_i x_i}{\sum w_i} $$ Where: - $w_i$ = weight of each value - $x_i$ = corresponding data value ### Example: Calculating Weighted Mean **Problem:** Calculate the average profit from truck shipments (in thousands of bags) from the USA to Canada, using the following data: | City | Weight ($w_i$) | Profit per 1000 bags ($x_i$) | |------------|----------------|-----------------------------| | Montreal | 64.0 | $15.00 | | Ottawa | 15.0 | $13.50 | | Toronto | 285.0 | $15.50 | | Vancouver | 228.0 | $12.00 | | Winnipeg | 45.0 | $14.00 | **Solution:** 1. Multiply each weight by its corresponding profit: - Montreal: $64.0 \times 15.00 = 960.0$ - Ottawa: $15.0 \times 13.50 = 202.5$ - Toronto: $285.0 \times 15.50 = 4417.5$ - Vancouver: $228.0 \times 12.00 = 2736.0$ - Winnipeg: $45.0 \times 14.00 = 630.0$ 2. Sum the weighted profits: $960.0 + 202.5 + 4417.5 + 2736.0 + 630.0 = 8946.0$ 3. Sum the weights: $64.0 + 15.0 + 285.0 + 228.0 + 45.0 = 637.0$ 4. Calculate the weighted mean: $$ \mu = \frac{8946.0}{637.0} \approx 14.04 $$ **Answer:** The average profit is **$14.04 per thousand bags**. --- ## The Center: Median The **median** is the middle value of a dataset when it is ordered from smallest to largest. ### Steps to Find the Median: 1. **Order the data** in ascending order. 2. If the dataset has: - **Odd** number of values: The median is the middle value. - **Even** number of values: The median is the average of the two middle values. The median is particularly useful when the dataset contains outliers, as it is less affected by extreme values compared to the mean. ### Example: Finding the Median **Dataset:** [12, 15, 18, 22, 25, 30, 35] - Ordered data: [12, 15, 18, 22, 25, 30, 35] - Number of values: 7 (odd) - Median: The 4th value = **22** **Dataset:** [10, 14, 18, 22, 26, 30] - Ordered data: [10, 14, 18, 22, 26, 30] - Number of values: 6 (even) - Median: Average of 18 and 22 = $\frac{18 + 22}{2} = 20$ --- ## The Center: Mode The **mode** is the most frequently occurring value in a dataset. ### Key Points: - A dataset can have: - **One mode** (unimodal) - **More than one mode** (multimodal) - **No mode** (if all values occur with the same frequency) - The mode is less commonly used than the mean or median. ### Example: Finding the Mode **Dataset:** [4, 5, 6, 6, 7, 8, 8, 8, 9] - The most frequent value is **8** (occurs 3 times). Do not confuse the mode with the mean or median. The mode is purely based on frequency, not position or calculation. --- ## Comparing Measures of Central Tendency The relationship between the **mean**, **median**, and **mode** can indicate the shape of a dataset's distribution: - **Symmetric Distribution:** Mean = Median = Mode - **Positively Skewed (Right-Tailed):** Mode < Median < Mean - **Negatively Skewed (Left-Tailed):** Mean < Median < Mode **Example:** - Dataset: [1, 2, 2, 3, 4, 5, 100] - Mean: $16.71$ - Median: $3$ - Mode: $2$ - Skew: Positively skewed (Mean > Median > Mode) --- ## The Spread: Range The **range** measures the spread of a dataset by calculating the difference between the largest and smallest values. ### Formula: $$ \text{Range} = \text{Largest Value} - \text{Smallest Value} $$ ### Example: Calculating Range **Dataset:** [5, 8, 12, 20, 25] - Largest value: 25 - Smallest value: 5 - Range: $25 - 5 = 20$ Sometimes, the range is reported as an interval, e.g., [5, 25], instead of the difference. --- ## Key Concept: Residuals **Residuals** are the differences between each data value and the group mean. - For a population: $x_i - \mu$ - For a sample: $x_i - \bar{x}$ Residuals are used to measure how far individual data points deviate from the mean. --- ## The Spread: Mean Absolute Deviation (MAD) The **Mean Absolute Deviation (MAD)** quantifies the average distance of each data point from the mean. ### Formula: - For a population: $$ \text{MAD} = \frac{\sum |x_i - \mu|}{N} $$ - For a sample: $$ \text{MAD} = \frac{\sum |x_i - \bar{x}|}{n} $$ ### Example: Calculating MAD **Dataset:** [2, 4, 6, 8, 10] (Population) 1. Mean: $\mu = \frac{2 + 4 + 6 + 8 + 10}{5} = 6$ 2. Residuals: $|2-6|, |4-6|, |6-6|, |8-6|, |10-6| = 4, 2, 0, 2, 4$ 3. MAD: $\frac{4 + 2 + 0 + 2 + 4}{5} = 2.4$ --- ## The Spread: Variance **Variance** measures the average squared deviation from the mean. It is a key concept in statistics, often used to calculate standard deviation. ### Formulas: - For a population: $$ \sigma^2 = \frac{\sum (x_i - \mu)^2}{N} $$ - For a sample: $$ s^2 = \frac{\sum (x_i - \bar{x})^2}{n - 1} $$ ### Shortcut Formula: - Population: $$ \sigma^2 = \frac{\sum x_i^2 - N \cdot \mu^2}{N} $$ - Sample: $$ s^2 = \frac{\sum x_i^2 - n \cdot \bar{x}^2}{n - 1} $$ Variance is always non-negative because squared deviations cannot be negative. --- ## Key Takeaways: 1. **Weighted Mean** accounts for the importance of different values. 2. **Median** is robust to outliers and splits the dataset into two equal halves. 3. **Mode** identifies the most frequent value(s) in a dataset. 4. **Range** provides a simple measure of spread but ignores intermediate values. 5. **Residuals** and **MAD** measure deviations from the mean. 6. **Variance** is a foundational concept for understanding data spread and variability. **Practice Problem:** 1. Calculate the weighted mean for the dataset: Weights = [2, 3, 5], Values = [10, 20, 30]. 2. Find the median of the dataset: [7, 3, 9, 1, 5]. 3. Compute the range for the dataset: [15, 22, 8, 19, 30].