One-Variable Data

One-Variable Data: Distributions and Measures of Center and Spread

SAT Math | Problem-Solving and Data Analysis | Foundation to Advanced

Data analysis is the process of organizing, summarizing, and interpreting information. On the SAT, you may need to calculate the mean, median, or range of a data set, understand how outliers affect statistical measures, read frequency tables and graphs, or compare how spread out two sets of data are.

1. Understanding One-Variable Data

One-variable data consists of observations about one characteristic.

For example, suppose seven students earn these test scores:

70, 75, 75, 80, 85, 90, 95

Only one variable is measured: the test score.

We can analyze this data to understand:

• A typical score.

• The lowest and highest scores.

• How much the scores vary.

• Whether some values are unusually far from the rest.

• How many students received particular scores.

The pattern of values is called a distribution.

A distribution describes which values occur and how frequently they occur.

Important terms

Observation: An individual value in a data set.

Frequency: The number of times a value occurs.

Distribution: The overall pattern of values and their frequencies.

Measure of center: A statistic describing a typical or central value.

Measure of spread: A statistic describing the variation among values.

Outlier: A value unusually far from the rest of the data.

On the SAT, data may be presented as an ordinary list, a frequency table, a dot plot, a histogram, or a box plot.

2. The Mean (Arithmetic Average)

The mean is the sum of all values divided by the number of values.

Formula:

Mean = Sum of all observations / Number of observations

For example:

4, 6, 8, 10, 12

Add the values:

4 + 6 + 8 + 10 + 12 = 40

Divide by 5:

Mean = 40 / 5 = 8

Mean = 8

Worked Example 1 — Calculating the mean

Difficulty: Easy

Five students read the following numbers of pages:

12, 18, 20, 25, 30

What is the mean number of pages?

Step 1: Add all values.

12 + 18 + 20 + 25 + 30 = 105

Step 2: Count the observations.

There are 5 students.

Step 3: Divide.

Mean = 105 / 5

Final answer: 21 pages

SAT shortcut: Total from mean

If you know the mean and the number of observations:

Total = Mean × Number of observations

For example, if 8 students have a mean score of 75:

Total score = 8 × 75 = 600

This is especially useful when a problem asks you to find a missing value.

3. Finding a Missing Value Using the Mean

Some SAT questions provide the mean but leave one data value unknown.

Worked Example 2 — Missing observation

Difficulty: Medium

The mean of five numbers is 14.

Four numbers are:

8, 12, 16, 19

What is the fifth number?

Step 1: Calculate the required total.

5 × 14 = 70

Step 2: Add the known values.

8 + 12 + 16 + 19 = 55

Step 3: Subtract.

70 − 55 = 15

Final answer: 15

Another important situation

Suppose a group of 6 values has a mean of 10.

A seventh value of 24 is added.

The original total is:

6 × 10 = 60

The new total is:

60 + 24 = 84

The new mean is:

84 / 7 = 12

New mean = 12

SAT tip: When values are added or removed, calculate the total first. Do not simply average the original mean and the new observation.

4. The Median

The median is the middle value of an ordered data set.

Always arrange the values from least to greatest before finding the median.

Odd number of observations

For:

3, 5, 8, 10, 14

There are 5 observations.

The middle value is 8.

Median = 8

Even number of observations

For:

4, 7, 11, 13, 18, 22

There are 6 observations.

The two middle values are 11 and 13.

Median = (11 + 13) / 2

Median = 12

Worked Example 3 — Median from unordered data

Difficulty: Medium

The following numbers represent seven temperatures:

16, 21, 13, 18, 24, 15, 19

What is the median?

Step 1: Arrange the values.

13, 15, 16, 18, 19, 21, 24

Step 2: Identify the middle observation.

The fourth value is 18.

Final answer: 18

Mean versus median

The mean uses every value in the calculation.

The median depends on the middle position.

An extreme value often affects the mean more than the median.

This distinction is frequently useful in SAT data interpretation.

5. The Mode

The mode is the value that occurs most frequently.

Consider:

2, 3, 3, 4, 5, 5, 5, 7

The value 5 occurs three times.

No other value occurs as often.

Mode = 5

Multiple modes

A data set can have more than one mode.

For example:

2, 2, 3, 4, 4, 5

Both 2 and 4 occur twice.

The data set is bimodal.

A data set can also have no unique mode if no value occurs more often than another.

Worked Example 4 — Identifying the mode

Difficulty: Easy

A survey records the number of books read by nine students:

1, 2, 2, 3, 3, 3, 4, 5, 5

What is the mode?

Count the frequencies:

1 occurs once.

2 occurs twice.

3 occurs three times.

4 occurs once.

5 occurs twice.

Final answer: 3

SAT tip: A frequency table or dot plot makes the mode easy to identify because it shows which value has the highest frequency.

6. The Range

The range measures the distance between the highest and lowest observations.

Formula:

Range = Maximum − Minimum

For example:

6, 9, 12, 18, 25

Maximum = 25

Minimum = 6

Range = 25 − 6

Range = 19

Worked Example 5 — Comparing ranges

Difficulty: Medium

Data Set A:

10, 12, 14, 16, 18

Data Set B:

4, 9, 14, 19, 24

Which data set has the greater range?

For A:

18 − 10 = 8

For B:

24 − 4 = 20

Final answer: Data Set B

Both sets have the same mean of 14 and median of 14, but Data Set B is more spread out.

Important limitation

Range uses only the smallest and largest values.

It does not describe how the remaining observations are distributed between those extremes.

7. Understanding Frequency Tables

A frequency table shows how many times each value occurs.

Consider:

This means:

• Three observations equal 2.

• Two observations equal 4.

• Five observations equal 6.

The total number of observations is:

3 + 2 + 5 = 10

Calculating mean from a frequency table

To find the mean:

1. Multiply each value by its frequency.

2. Add the products.

3. Divide by the total frequency.

Formula:

Mean = Σ(value × frequency) / Σfrequency

Worked Example 6 — Frequency table mean

Difficulty: Medium

Find the mean.

Step 1: Multiply each value by its frequency.

2 × 3 = 6

5 × 2 = 10

8 × 1 = 8

Step 2: Add the products.

6 + 10 + 8 = 24

Step 3: Find the total frequency.

3 + 2 + 1 = 6

Step 4: Divide.

24 / 6 = 4

Final answer: 4

Finding median from a frequency table

Expand the observations mentally in order, or use cumulative frequencies to locate the middle position.

For the table above, the six observations are:

2, 2, 2, 5, 5, 8

The middle values are the third and fourth:

2 and 5

Median = (2 + 5) / 2 = 3.5

SAT tip: Frequency tables often allow you to solve problems without writing every observation individually.

8. Reading Dot Plots

A dot plot represents observations using dots placed above numerical values.

Each dot represents one observation.

For example, imagine a dot plot showing the following frequencies:

The total number of observations is:

2 + 4 + 3 + 1 = 10

Finding the median

The ordered observations are:

1, 1, 2, 2, 2, 2, 3, 3, 3, 4

There are 10 observations.

The fifth and sixth values are both 2.

Median = 2

Finding the mode

The value 2 has the greatest number of dots.

Mode = 2

Finding the range

Maximum = 4

Minimum = 1

Range = 3

Worked Example 7 — Comparing dot plots

Difficulty: Medium

Two dot plots represent the following data.

Data Set A:

8, 9, 10, 10, 11, 12

Data Set B:

2, 6, 10, 10, 14, 18

Which data set is more spread out?

Data Set A:

Range = 12 − 8 = 4

Data Set B:

Range = 18 − 2 = 16

Both have mean 10 and median 10.

However, Data Set B has a much greater spread.

Final answer: Data Set B

9. Understanding Histograms

A histogram displays frequencies for intervals of numerical data.

For example:

The tallest bar represents the interval with the greatest frequency.

In this example, the 10–19 minute interval has the most observations.

Reading a histogram

A histogram can show:

• How many observations fall within each interval.

• Which interval contains the most values.

• Where values are concentrated.

• How widely the values are distributed.

• Whether the distribution appears symmetric or skewed.

Worked Example 8 — Histogram frequency

Difficulty: Easy

The number of students in each time interval is:

How many students studied for at least 20 minutes?

Count the students in intervals beginning at 20 minutes.

6 + 2 = 8

Final answer: 8 students

Important limitation

A histogram groups values into intervals.

Unless more information is supplied, the exact individual observations are unknown.

For example, if five students studied between 20 and 29 minutes, we cannot determine their exact times from the histogram alone.

SAT tip: Read interval boundaries carefully. Do not assume the middle of an interval is the exact value of every observation.

10. Understanding Box Plots

A box plot summarizes a numerical distribution using five important values.

These are called the five-number summary:

1. Minimum

2. First quartile (Q1)

3. Median (Q2)

4. Third quartile (Q3)

5. Maximum

First quartile

Q1 marks approximately the 25th percentile of the data.

Median

Q2 is the 50th percentile.

Third quartile

Q3 marks approximately the 75th percentile.

Interquartile range

The interquartile range, or IQR, measures the spread of the middle half of the data.

Formula:

IQR = Q3 − Q1

Worked Example 9 — Interquartile range

Difficulty: Medium

A box plot has:

Minimum = 2

Q1 = 6

Median = 10

Q3 = 15

Maximum = 22

What is the interquartile range?

Step 1: Identify Q1 and Q3.

Q1 = 6

Q3 = 15

Step 2: Subtract.

IQR = 15 − 6

Final answer: 9

What each part of a box plot means

The box extends from Q1 to Q3.

The line inside the box shows the median.

The whiskers extend toward the smallest and largest non-outlier values in box plots that display outliers separately; in simpler SAT-style box plots, they may be drawn to the minimum and maximum.

Always follow the given graph's labels and conventions.

Comparing box plots

A larger IQR indicates that the middle 50% of the observations are more spread out.

A larger range indicates a greater overall distance between the minimum and maximum observations.

These are not the same measure.

11. Understanding Standard Deviation

Standard deviation describes how far observations typically vary from their mean.

A low standard deviation indicates that values are relatively close to the mean.

A high standard deviation indicates that values are more spread out from the mean.

Example

Data Set A:

9, 10, 11

Data Set B:

1, 10, 19

Both have mean 10.

However, Data Set A is tightly clustered near 10.

Data Set B is much more spread out.

Therefore:

Data Set B has the greater standard deviation.

Important SAT requirement

For the SAT, you should understand how to compare and interpret standard deviation.

You are generally not required to calculate it using the standard deviation formula.

When all values increase by the same amount

Adding 5 to every observation:

• Increases the mean by 5.

• Increases the median by 5.

• Leaves the range unchanged.

• Leaves the standard deviation unchanged.

When all values are multiplied by a positive factor

Multiplying every value by 3:

• Multiplies the mean by 3.

• Multiplies the median by 3.

• Multiplies the range by 3.

• Multiplies standard deviation by 3.

Worked Example 10 — Comparing standard deviations

Difficulty: Medium

Two data sets contain:

Data Set A:

18, 19, 20, 21, 22

Data Set B:

4, 12, 20, 28, 36

Both have mean 20.

Which has the greater standard deviation?

Data Set A is concentrated near 20.

Data Set B contains values much farther from 20.

Final answer: Data Set B

SAT tip: When two distributions have the same mean, compare how far their observations are from that mean.

12. Outliers and Their Effects

An outlier is an unusually high or low observation compared with the rest of the data.

Consider:

5, 7, 8, 8, 9

Mean:

(5 + 7 + 8 + 8 + 9) / 5 = 7.4

Median = 8

Now replace 9 with 90.

The new data set is:

5, 7, 8, 8, 90

New mean:

(5 + 7 + 8 + 8 + 90) / 5 = 23.6

New median = 8

What changed?

The mean increased substantially.

The median stayed the same.

The range increased.

The spread increased.

Mean versus median with outliers

The mean is sensitive to extreme values.

The median is generally more resistant to extreme values.

That is why the median is often a better description of a typical value in a highly skewed distribution.

Worked Example 11 — Outlier effect

Difficulty: Medium

A data set contains:

4, 6, 8, 10, 12

The largest value is replaced by 42.

Which measure remains unchanged?

The original median is 8.

The new data set is:

4, 6, 8, 10, 42

The middle observation is still 8.

Final answer: Median

The mean and range both increase.

Important: Removing an outlier does not guarantee that every measure of center remains unchanged. Always examine the actual data.

13. Comparing Data Distributions

The SAT may present two data sets and ask which has:

• The greater mean.

• The greater median.

• The larger range.

• The larger IQR.

• The greater standard deviation.

• More observations above or below a given value.

Worked Example 12 — Same center, different spread

Difficulty: Medium

Data Set A:

6, 8, 10, 12, 14

Data Set B:

2, 6, 10, 14, 18

Find the mean and range of each data set.

For A:

Total = 50

Mean = 50/5 = 10

Range = 14 − 6 = 8

For B:

Total = 50

Mean = 50/5 = 10

Range = 18 − 2 = 16

Conclusion: Both data sets have the same mean, but Data Set B has the greater range.

Comparing center and variability

A distribution can have:

• A higher mean but a smaller spread.

• A lower mean but a larger spread.

• The same mean and different standard deviations.

• The same median and different ranges.

Never assume that the data set with the greater average also has greater variability.

14. Interpreting the Shape of a Distribution

A distribution's shape describes how the observations are arranged.

Approximately symmetric distributions

In a symmetric distribution, the left and right sides are approximately balanced around the center.

The mean and median are often close.

Right-skewed distributions

A right-skewed distribution has a longer tail toward large values.

Example:

4, 5, 6, 6, 7, 8, 46

The unusually high observation pulls the mean upward.

The mean is greater than the median.

Left-skewed distributions

A left-skewed distribution has a longer tail toward small values.

A very low observation can pull the mean below the median.

Worked Example 13 — Skewed data

Difficulty: Medium

A group of incomes, in thousands of dollars, is:

30, 32, 33, 35, 36, 38, 146

Which measure of center better represents a typical income?

The median is 35.

The mean is:

(30 + 32 + 33 + 35 + 36 + 38 + 146)/7

= 350/7

= 50

The high income of 146 pulls the mean above most observations.

Final answer: The median, 35 thousand dollars.

The median is more representative of the typical income in this data set.

15. Combined Data Sets and Weighted Means

Some SAT questions provide separate groups and their mean values, then ask for the mean of the combined group.

You cannot simply average the group means unless the groups have the same number of observations.

Formula

Combined mean = Total sum of all groups / Total number of observations

For two groups:

Combined mean = (n1 × mean1 + n2 × mean2) / (n1 + n2)

Worked Example 14 — Combined mean

Difficulty: Hard

One class has 8 students with a mean quiz score of 14.

Another class has 12 students with a mean quiz score of 19.

What is the mean score of all 20 students?

Step 1: Calculate the first group's total.

8 × 14 = 112

Step 2: Calculate the second group's total.

12 × 19 = 228

Step 3: Add both totals.

112 + 228 = 340

Step 4: Divide by the total number of students.

340 / 20 = 17

Final answer: 17

Why averaging the means is incorrect

(14 + 19)/2 = 16.5

This gives equal weight to both groups even though they contain different numbers of students.

The correct combined mean is 17.

SAT tip: Multiply each group mean by its group size before combining the data.

16. Advanced SAT Data Questions

Worked Example 15 — Finding an added observation

Difficulty: Hard

A data set contains 6 values with a mean of 15.

One additional value is added, increasing the mean to 17.

What is the added value?

Step 1: Find the original total.

6 × 15 = 90

Step 2: Find the new total.

7 × 17 = 119

Step 3: Subtract.

119 − 90 = 29

Final answer: 29

Worked Example 16 — Unknown frequency

Difficulty: Hard

A frequency table contains:

The mean is 16.

What is n?

Step 1: Write the total sum.

10(2) + 15(4) + 20n

= 80 + 20n

Step 2: Write the total frequency.

2 + 4 + n = 6 + n

Step 3: Form the mean equation.

(80 + 20n)/(6 + n) = 16

Step 4: Multiply.

80 + 20n = 96 + 16n

Step 5: Rearrange.

4n = 16

Final answer: n = 4

Worked Example 17 — Changing every observation

Difficulty: Hard

A data set has mean 8 and standard deviation 3.

A new data set is created by multiplying every value by 2 and then adding 5.

What are the new mean and standard deviation?

Step 1: Transform the mean.

New mean = 2(8) + 5 = 21

Step 2: Transform standard deviation.

Multiplying by 2 doubles the spread.

Adding 5 does not change the spread.

New standard deviation = 2(3) = 6

Final answer: Mean 21; standard deviation 6

17. SAT Strategies and Common Mistakes

Strategy 1 — Order data before finding the median

The median depends on position, so arrange the observations in numerical order.

Strategy 2 — Use totals to solve mean questions

Mean × Number of observations = Total.

This is useful for missing values, combined means, and changes in data sets.

Strategy 3 — Read graph frequencies carefully

Each dot represents one observation.

In a frequency table, each frequency indicates repeated occurrences.

In a histogram, values are grouped into intervals.

Strategy 4 — Distinguish center from spread

Mean and median describe center.

Range, IQR, and standard deviation describe spread.

Strategy 5 — Check outlier effects

An extreme observation may strongly change the mean and range while having less influence on the median.

Strategy 6 — Interpret standard deviation

Values close to the mean suggest a smaller standard deviation.

Values far from the mean suggest a larger standard deviation.

Strategy 7 — Compare quartiles correctly

The interquartile range is Q3 − Q1.

Do not confuse it with the full range.

Strategy 8 — Use appropriate statistics in context

The mean uses every numerical observation.

The median may better represent a typical value when extreme observations distort the average.

Common Mistake 1 — Forgetting frequencies

When calculating mean from a frequency table, multiply each value by its frequency.

Common Mistake 2 — Using the wrong median

For an even number of observations, average the two middle values.

Common Mistake 3 — Confusing range and IQR

Range = Maximum − Minimum.

IQR = Q3 − Q1.

Common Mistake 4 — Assuming the mean must be one of the observations

The mean can fall between observations or be a number not present in the original data.

Common Mistake 5 — Averaging group means directly

For groups of different sizes, use a weighted combined mean.

Common Mistake 6 — Assuming the highest mean implies the greatest spread

Center and spread are different characteristics.

Common Mistake 7 — Treating histogram intervals as exact observations

A histogram does not usually reveal the precise values of every observation.

Common Mistake 8 — Ignoring the effects of outliers

An extreme value may substantially affect the mean and standard deviation.

Common Mistake 9 — Calculating standard deviation unnecessarily

For SAT questions, focus on interpreting and comparing how spread out distributions are.

Common Mistake 10 — Confusing sample size with the range

The number of observations measures how much data was collected.

The range measures the distance between the maximum and minimum values.

18. Check Your Understanding

Question 1 — Mean

Find the mean of:

5, 8, 11, 14, 17

A. 9

B. 10

C. 11

D. 12

Correct answer: C — 11

Total = 55.

Mean = 55/5 = 11.

Question 2 — Median

Find the median of:

4, 7, 9, 12, 15, 20

A. 9

B. 10.5

C. 12

D. 13.5

Correct answer: B — 10.5

The two middle values are 9 and 12.

Median = (9 + 12)/2 = 10.5.

Question 3 — Frequency table

A frequency table shows:

What is the mean?

A. 3

B. 4

C. 5

D. 6

Correct answer: B — 4

Weighted sum = 2(3) + 5(2) + 8(1) = 24.

Total frequency = 6.

Mean = 24/6 = 4.

Question 4 — Box plot

A box plot has Q1 = 8 and Q3 = 21.

What is the interquartile range?

Correct answer: 13

IQR = 21 − 8 = 13.

Question 5 — Standard deviation

Data Set A contains:

9, 10, 11

Data Set B contains:

1, 10, 19

Which data set has the greater standard deviation?

Correct answer: Data Set B

Both means are 10, but Data Set B's observations are farther from the mean.

19. Lesson Summary

One-variable data describes observations about one characteristic.

A distribution shows the values and frequencies in a data set.

The mean is the sum of observations divided by the number of observations.

The median is the middle value after ordering the data.

The mode is the most frequent value.

The range is the maximum value minus the minimum value.

Frequency tables and dot plots show repeated observations.

Histograms group numerical observations into intervals.

Box plots display the five-number summary.

The interquartile range describes the spread of the middle half of the data.

Standard deviation describes variation around the mean.

Outliers can affect measures of center and spread.

Combined means require accounting for the sizes of the individual groups.

Comparing distributions requires examining both their centers and their variability.

You are now ready to complete One-Variable Data practice questions in Baloo Prep.