Mean, Median, and Mode

To master this section, you must be comfortable with three primary measures of center and the specific ways the SAT presents them.

1. The Arithmetic Mean

The mean is the "average" you are most familiar with. It is calculated by summing all the values in a data set and dividing by the total number of values.

The Formula: $\text{Mean} (\bar{x}) = \frac{\sum x}{n} = \frac{\text{Sum of values}}{\text{Number of values}}$

The SAT "Sum" Trick: Most students focus on the mean itself, but the SAT often gives you the mean and asks for a missing value. In these cases, the Sum Formula is much more useful: $\text{Sum} = \text{Mean} \times \text{Number of values}$ If you know the average of 5 tests is 90, you immediately know the total points earned is $90 \times 5 = 450$. This is the most common "move" on mean-related word problems.

2. The Median

The median is the middle value of a data set when the numbers are arranged in ascending or descending order.

• If $n$ is odd: The median is the exact middle number. Its position can be found using $\frac{n+1}{2}$.
• If $n$ is even: The median is the average of the two middle numbers.

When it appears: The SAT loves to ask for the median from a frequency table or dot plot. You must be careful not to pick the middle number in the "frequency" column, but rather the value that corresponds to the middle position of the total count.

3. The Mode

The mode is the value that appears most frequently in a data set. A set can have one mode, more than one mode (bimodal/multimodal), or no mode at all. On the SAT, mode is usually the simplest concept tested, often appearing in questions asking you to identify the most common response in a survey.

4. Weighted Averages

A weighted average occurs when different groups contribute differently to the final mean. For example, if Class A has 10 students with an average of 80 and Class B has 20 students with an average of 90, the total average is NOT 85.

The Formula: $\text{Weighted Mean} = \frac{(n_1 \times \bar{x}_1) + (n_2 \times \bar{x}_2)}{n_1 + n_2}$ In our example: $\frac{(10 \times 80) + (20 \times 90)}{10 + 20} = \frac{800 + 1800}{30} \approx 86.7$.

5. Mean vs. Median (Shape of Data)

The SAT frequently asks how an outlier (a value much higher or lower than the rest) affects the mean and median.

• Mean: Highly sensitive to outliers. A very high value will pull the mean up; a very low value will pull it down.
• Median: Resistant to outliers. Because it only cares about the middle position, an extreme value at the end of the list rarely changes the median significantly.

Important Note: None of these formulas are provided on the SAT Reference Sheet. You must memorize them and, more importantly, understand how to manipulate them.

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1. Use the Desmos Calculator

The Digital SAT includes an integrated Desmos calculator. For any question providing a list of numbers, use the mean() and median() functions.

• Type L = [1, 2, 3, 4, 5] to create a list.
• Type mean(L) or median(L) to get instant results. This eliminates calculation errors and saves precious seconds.

2. The "Balance" Concept

Think of the mean as the balance point of a seesaw. If you add a number greater than the current mean, the mean must increase. If you add a number smaller than the mean, the mean must decrease. If you add a number exactly equal to the mean, the mean stays the same. This conceptual understanding often allows you to solve "impact" questions without doing any math.

3. Frequency Table Navigation

When given a frequency table, always calculate the cumulative frequency to find the median. If there are 51 total items, the median is the $26^{th}$ item ($\frac{51+1}{2} = 26$). Count through the frequencies until you hit the $26^{th}$ value.

4. Watch for "Must be True" vs. "Could be True"

Data analysis questions often use this phrasing. If a question asks what must be true about the median when a value is added, test the most extreme scenarios. If the conclusion holds in all cases, it's the correct answer.

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