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Center, Shape, &
Spread of Data

Master the core statistical measures: mean, median, standard deviation, and how data distributions reveal patterns on the SAT.

SAT Math

What Is the Center of Data?

The center of a data set is a single value that represents a "typical" observation. On the SAT, you will see three measures of center:

Mean
The arithmetic average: \(\bar{x} = \frac{\sum x_i}{n}\)
Median
The middle value when data is ordered from least to greatest
Mode
The value that appears most frequently in a data set

The Shape of a Distribution

The shape of a data set describes how values are distributed. Understanding shape tells you the relationship between mean and median.

Left Skewed Symmetric Right Skewed

Understanding Spread

Spread measures how much variation exists in a data set. Two distributions can have the same center but very different spreads.

Range
\(\text{max} - \text{min}\): simplest measure, but sensitive to outliers
IQR
\(Q_3 - Q_1\): the range of the middle 50% of data
Std Dev (\(\sigma\))
Average distance of each data point from the mean

The Mean Formula

$$\bar{x} = \frac{x_1 + x_2 + \cdots + x_n}{n} = \frac{\sum_{i=1}^{n} x_i}{n}$$

1
Add all values. Sum every data point in the set.
2
Count the values. Determine \(n\), the total number of data points.
3
Divide. The quotient is the mean.

Visualizing Standard Deviation

Mean

\(\text{Large } \sigma\)

\(\text{Small } \sigma\)

\(\text{wider spread} = \text{larger } \sigma\)

Mean vs. Median: Key Distinction

Mean \(\bar{x}\) Median
Calculation Sum divided by count Middle value (ordered)
Affected by outliers? Yes, strongly No, resistant
Best for Symmetric data Skewed data
Skewed right \(\text{Mean} > \text{Median}\) \(\text{Median} < \text{Mean}\)
Skewed left \(\text{Mean} < \text{Median}\) \(\text{Median} > \text{Mean}\)
Symmetric Mean ≈ Median

How to Analyze a Data Set

1
Identify the shape. Look at histograms or dot plots: is the distribution symmetric, left-skewed, or right-skewed?
2
Choose the right center. Use mean for symmetric data, median for skewed data or when outliers are present.
3
Measure the spread. Use IQR for skewed data, standard deviation for symmetric data.
4
Check for outliers. A value is an outlier if it falls below \(Q_1 - 1.5 \cdot \text{IQR}\) or above \(Q_3 + 1.5 \cdot \text{IQR}\).
5
Interpret in context. Always relate your answer back to the real-world scenario described in the problem.

Example 1: Finding a Missing Value

A student scored 78, 85, 92, 88 on four tests. What score does she need on the fifth test to have a mean of 87?

1
Use the mean formula: \(\bar{x} = \frac{\text{Sum}}{n}\), so \(\text{Sum} = \bar{x} \times n = 87 \times 5 = 435\)
2
Find the current sum: \(78 + 85 + 92 + 88 = 343\)
3
Subtract: \(435 - 343 = 92\)

Example 2: Comparing Two Data Sets

Two classes took the same quiz. Which class had more consistent scores?

\(\text{Class A } (\sigma = 5.2)\)

\(\text{Class B } (\sigma = 11.8)\)

Example 3: Outlier Effects

Data set: 10, 12, 11, 13, 12, 11, 10, 50. How does the outlier (50) affect center and spread?

With Outlier (50) Without Outlier
Mean 16.1 11.3
Median 11.5 11
Range 40 3
Std Dev 12.8 1.0

Reading a Box Plot

10 20 30 40 50 60 Min

\(Q_1\)

 Median

\(Q_3\)

 Max

\(\text{IQR} = Q_3 - Q_1\)

How Changes Affect Statistics

The SAT loves asking: "If a value is added or removed, what happens to the mean/median/standard deviation?"

1
Adding a value equal to the mean: mean stays the same, standard deviation decreases
2
Adding a value far from the mean: mean shifts toward that value, standard deviation increases
3
Removing an outlier: mean moves toward the bulk of data, standard deviation decreases
4
Adding a constant to every value: mean and median increase by that constant, standard deviation stays the same

The Weighted Mean

When groups have different sizes, you cannot simply average the group means. You must use a weighted mean:

$$\bar{x}_w = \frac{n_1 \bar{x}_1 + n_2 \bar{x}_2}{n_1 + n_2}$$

Example: Group A has 20 students with mean score 80. Group B has 30 students with mean score 90. What is the overall mean?

Common Mistakes

✕ Averaging Averages
You cannot average group means unless groups are equal size. Use the weighted mean formula instead.
✕ Confusing Skew Direction
The skew is named for the tail direction, not the peak. Right skew = tail goes right = \(\text{mean} > \text{median}\).
✕ Ignoring Outlier Effects
Outliers pull the mean and inflate standard deviation, but barely affect the median or IQR.
✕ Mixing Up IQR and Range
\(\text{Range} = \text{max} - \text{min}\). IQR = \(Q_3 - Q_1\). They measure different things. IQR is resistant to outliers.

Summary of Key Measures

Measure What It Tells You Resistant? Best For
Mean Average value No Symmetric data
Median Middle value Yes Skewed / outliers
Range Total spread No Quick comparison
IQR Middle 50% spread Yes Skewed / outliers
Std Dev (\(\sigma\)) Avg distance from mean No Symmetric data

Question 1 of 5

A data set has a mean of 50 and a median of 42. Which of the following best describes the shape of this distribution?

Question 2 of 5

The five-number summary for a data set is: 12, 25, 34, 47, 60. What is the IQR?

Question 3 of 5

A class of 10 students has a mean test score of 82. If the highest score of 100 is removed, what is the new mean of the remaining 9 students? (Round to the nearest tenth.)

Question 4 of 5

If every value in a data set is multiplied by 3, which of the following is true?

Question 5 of 5

Group A: 15 students, mean score 70. Group B: 25 students, mean score 86. What is the combined mean of all 40 students?