Finding the Median (Odd Number of Values)

Find the median of: $12, 5, 8, 3, 9$

Arrange in order from least to greatest: $3, 5, 8, 9, 12$ = 5 values in order

Count the values: There are 5 values (odd number) = Look for the middle value

Find the middle position: Position = $(5 + 1) \div 2 = 3$ = 3rd value is the median

Identify the median: $3, 5, \boxed{8}, 9, 12$ = Median = $8$

Answer: The median is $8$

Finding the Median (Even Number of Values)

Find the median of: $4, 7, 10, 15$

Check if already in order: $4, 7, 10, 15$ - Yes, already ordered = 4 values in order

Count the values: There are 4 values (even number) = Need average of two middle values

Find the two middle values: $4, \boxed{7, 10}, 15$ = Middle values: $7$ and $10$

Calculate their average: $(7 + 10) \div 2 = 17 \div 2 = 8.5$ = Median = $8.5$

Answer: The median is $8.5$

Finding the Mode

Find the mode of: $3, 7, 3, 9, 3, 5, 7$

Organize and count each value: $3$ appears 3 times, $5$ appears 1 time, $7$ appears 2 times, $9$ appears 1 time = Counted all frequencies

Find the highest frequency: $3$ appears most often (3 times) = $3$ is the mode

Verify no ties: No other value appears 3 times = Single mode

Answer: The mode is $3$

Data Set with Multiple Modes

Find the mode of: $2, 4, 4, 6, 6, 8$

Count each value: $2$: 1 time, $4$: 2 times, $6$: 2 times, $8$: 1 time = Counted all frequencies

Find the highest frequency: Both $4$ and $6$ appear 2 times each = Tied for most frequent

Identify all modes: When two values tie, we have two modes = Bimodal data set

Answer: The modes are $4$ and $6$ (this is called bimodal)

Comparing Mean, Median, and Mode

A basketball player scored these points in 5 games: $8, 10, 10, 12, 45$. Find all three measures.

Find the mean: $(8 + 10 + 10 + 12 + 45) \div 5 = 85 \div 5 = 17$ = Mean = $17$

Find the median: Already ordered, middle value is $\boxed{10}$ = Median = $10$

Find the mode: $10$ appears twice, all others once = Mode = $10$

Analyze the difference: The 45-point game pulls the mean up, but median and mode show typical performance = Median/mode better represent typical games

Answer: Mean = $17$, Median = $10$, Mode = $10$. The median and mode ($10$) better represent this player's typical game.

Mistake: Forgetting to order the data before finding the median

Why: Without ordering, you might pick a value that's not actually in the middle.

Correct: ALWAYS sort from least to greatest first: $5, 2, 8, 1 \rightarrow 1, 2, 5, 8$

Mistake: Saying there's no mode when all values appear once

Why: This is actually correct! A data set can have no mode.

Correct: If every value appears the same number of times, there is no mode.

Mistake: Confusing median position with median value

Why: In $1, 3, 5, 7, 9$, the median is at position 3, but the median VALUE is $5$.

Correct: First find the position, then identify the value at that position.

Mistake: For even data sets, picking one of the middle values instead of their average

Why: With even counts, the median falls between two values.

Correct: For $4, 6, 8, 10$: median = $(6 + 8) \div 2 = 7$, not just $6$ or $8$.

Real Estate Prices

Home prices are often reported as median rather than mean because a few luxury homes can drastically increase the average.

In a neighborhood, home prices are: 200000, 220000, 250000, 280000, and 1500000 dollars. The mean is 490000 dollars, but the median is 250000 dollars - much more representative of typical homes.

Retail Inventory

Stores use mode to determine which products to stock more of.

T-shirt sales by size: S(12), M(28), L(28), XL(15). The modes are M and L - the store should order more medium and large shirts.

Test Scores

Teachers use median to understand class performance without outliers affecting the result.

Class scores: 45, 72, 75, 78, 80, 82, 85, 88, 92. The median is 80, showing where the middle student performed.

Q: Can a number be both the median AND the mode?

A: Yes! In $3, 5, 5, 5, 7$, the number $5$ is both the median (middle value) and the mode (appears most often).

Q: What if no number repeats?

A: If every value appears exactly once, there is no mode. For example, $2, 4, 6, 8$ has no mode.

Q: Why not always use the mean?

A: The mean can be pulled by extreme values. If 4 students score 80 and one scores 5, the mean is 65, but the median (80) better represents typical performance.