One-variable data: Distributions and measures of center and spread

One-variable data analysis looks at a single set of numbers to find the average, see how spread out the values are, and understand the overall shape of the data.

Basic Measures of Center and Spread

• Mean: The total sum of all numbers divided by how many numbers there are (the average).

  Example: For $\{2, 4, 6, 8, 10\}$

  Mean $= \frac{30}{5} = 6$.

  
  

• Median: The middle number when all values are listed in order from smallest to largest.

  Example: For $\{3, 5, 8, 12, 14\}$, the middle number is $8$.

  
  

• Mode: The number that shows up most often in the dataset.

  Example: For $\{2, 3, 3, 5, 7\}$, the mode is $3$.

  
  

• Range: The difference between the highest value and the lowest value.

  Example: For $\{4, 7, 10, 25\}$, Range $= 25 - 4 = 21$.

  
  

• Standard Deviation: A measure of how close or far the numbers are spread out from the mean.

  Example: $\{10, 10, 10\}$ has a standard deviation of $0$ (no spread), while $\{2, 10, 18\}$ has a large standard deviation (widely spread).

  
  

The Effect of Outliers

An outlier is an extreme value that is much higher or much lower than the rest of the numbers in a dataset.

• Effect on the Mean: Heavily affected. Outliers pull the mean toward the extreme value, making it less representative of the typical data.

  Example: For dataset $\{10, 12, 14, 14, 15\}$, mean $= 13$. Adding an outlier of $100$ gives $\{10, 12, 14, 14, 15, 100\}$, spiking the new mean to $27.5$.

• Effect on the Median: Barely affected. Because the median only looks at the middle position, extreme values at the ends do not change it much.

  Example: For $\{10, 12, 14, 14, 15\}$, median $= 14$. With the outlier $100$, the set becomes $\{10, 12, 14, 14, 15, 100\}$, making the new median $\frac{14 + 14}{2} = 14$.

• Effect on the Range: Heavily affected. Since range depends directly on the maximum and minimum values, an extreme outlier increases the range dramatically.

  Example: Original range for $\{10, 12, 14, 14, 15\}$ is $15 - 10 = 5$. With the outlier $100$, the new range becomes $100 - 10 = 90$.

1. Pick the median when a dataset has extreme outliers.

   Example: House prices in a neighborhood are $\$200\text{k}, \$210\text{k}, \$220\text{k}$, and one mansion worth $\$2,000\text{k}$.

   • The mansion pulls the mean up to $\$657.5\text{k}$, which is misleading.

   • The median ($\$215\text{k}$) accurately represents typical home prices.

2. Understand how adding or multiplying values changes center and spread.

   Example: Adding $\$5$ to every item price vs. doubling every item price.

   • Adding $\$5$: Mean and median increase by $\$5$, but range and standard deviation stay the same.

   • Doubling prices: Both center (mean/median) and spread (range/standard deviation) double.

3. Compare standard deviation by looking at how far numbers sit from the center.

   Example: Set $A = \{4, 5, 5, 6\}$ vs. Set $B = \{1, 5, 5, 9\}$.

   • Both sets have a mean of $5$.

   • Set $B$ has a larger standard deviation because $1$ and $9$ sit much farther from $5$ than $4$ and $6$.