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SAT Data Analysis – Quick Review

satmath80017 mins
SAT Data Analysis | SATMath800
DIGITAL SAT · DATA ANALYSIS
SATMath800 Lesson

Data
Analysis

Read the data. See the pattern. Solve the problem.

Master ratios, rates, percentages, statistics, distributions, graphs, models, probability, sampling, and study design—one clear idea at a time.

Sample Data
Group Mean
A 72
B 81
C 78
D 86
Distribution
Scatterplot
01

The SAT Data Detective

Start by understanding what the data is saying.

What Is Data Analysis?

SAT Data Analysis questions are not usually about doing complicated mathematics. They are about extracting the right information, choosing the right mathematical idea, and interpreting the result.

READ
What information is shown?
INTERPRET
What does it mean?
DECIDE
What must I calculate?
READ INTERPRET CALCULATE DECIDE
The SAT Data Analysis mindset
SATMath800 principle: Do not calculate information the table or graph has already given you.
02

Ratios, Rates & Percentages

Translate quantities into useful relationships.

Ratios: Parts, Groups & Proportions

If red objects and blue objects are in the ratio \(3:5\), there are \(3+5=8\) total parts.

1
Total parts: \(3+5=8\)
2
If there are \(40\) objects, one part is \(40\div8=5\).
3
Red objects: \(3(5)=15\).
3 : 5 3 parts 5 parts
Three parts to five parts—not three objects to five objects.
Ratio recipe: Ratio → total parts → value per part → desired quantity.

Rates & Units

A rate compares quantities with different units. On the SAT, the units themselves can guide your setup.

\[ 60\frac{\text{miles}}{\text{hour}} \times 2\text{ hours} = 120\text{ miles} \]
Unit Detective: If you want miles, make sure every unwanted “hour” cancels.

Example: A machine produces \(18\) parts per minute. At the same rate, how many parts does it produce in \(5\) minutes?

\[ 18\frac{\text{parts}}{\text{minute}} \times 5\text{ minutes} = 90\text{ parts} \]

Percentages: The SAT’s Favorite Language

Percent of a quantity

\[ 20\%\text{ of }80 = 0.20(80) = 16 \]

Percent change

\[ \frac{ \text{new}-\text{original} }{ \text{original} } \times100\% \]
The 100 Trick: When the original quantity is awkward, imagine it as \(100\). Then percentage changes often become much easier to see.

Example: A quantity increases from \(80\) to \(100\).

\[ \frac{100-80}{80}\times100\% = 25\% \]
⚠
Common trap: A change from \(20\%\) to \(25\%\) is an increase of 5 percentage points, but the relative percent increase is \(25\%\).
03

Understanding Distributions

Mean, median, spread, outliers, and standard deviation.

Mean: The Balance Point

The mean is the arithmetic average, but for SAT problems the most useful relationship is often:

\[ \boxed{ \text{Total} = \text{Mean} \times \text{Number of Values} } \]

Example: The mean of \(8\) numbers is \(14\). What is their total?

\[ 8(14)=112 \]
LOWER HIGHER mean = balance point
Think of the mean as a balance point for the data.

Finding a Missing Value

Suppose the mean of \(8,\ 11,\ 14,\ x\) is \(12\).

1
There are \(4\) values, so the required total is \[ 4(12)=48. \]
2
The known values total \[ 8+11+14=33. \]
3
Therefore, \[ x=48-33=\boxed{15}. \]
Shortcut: Find the required total first. Then subtract what you already know.

Median: The Middle Matters

The median is the middle value after the data are placed in numerical order.

Odd number of values

\[ 4,\ 7,\ \boxed{9},\ 12,\ 15 \]

Median \(=9\).

Even number of values

\[ 4,\ \boxed{7,\ 9},\ 12 \]

The two middle values are \(7\) and \(9\).

\[ \text{Median} = \frac{7+9}{2} = \boxed{8} \]
Put the values in order first 4 7 9 12 15 middle value = median
For an even number of observations, average the two middle values.

Mean vs. Median: Outliers Change the Story

\[ 10,\ 11,\ 12,\ 13,\ 14,\ 80 \]

The \(80\) is far from the rest of the data. It pulls the mean upward, while the median remains much more stable.

Mean
Sensitive to extreme values.
Median
Much less affected by extreme values.
outlier most observations
One extreme value can have a large effect on the mean.

Range & Spread

\[ \boxed{ \text{Range} = \text{Maximum} – \text{Minimum} } \]
Data A
\(48,\ 49,\ 50,\ 51,\ 52\)
Range: \[ 52-48=4 \]
Data B
\(20,\ 35,\ 50,\ 65,\ 80\)
Range: \[ 80-20=60 \]

Both data sets have mean \(50\), but Data B is much more spread out.

Standard Deviation: How Spread Out?

For the SAT, the important idea is interpretation: standard deviation measures how spread out the data are around the mean.

mean
Small standard deviation: clustered data
mean
Large standard deviation: spread-out data
Same mean + narrower distribution = smaller standard deviation.
Same mean + wider distribution = larger standard deviation.
04

Reading Graphs & Tables

Turn visual information into mathematics.

Frequency & Relative Frequency

Frequency tells you how many observations are in a category or interval.

Relative frequency tells you what fraction or percentage of the total that represents.

\[ \text{Relative frequency} = \frac{\text{frequency}}{\text{total}} \]

Example: \(24\) of \(80\) students prefer option A.

\[ \frac{24}{80} = 0.30 = 30\% \]

Dot Plots: Every Dot Counts

1 2 3 4 5 6 7 Each dot = one observation
From a dot plot, you can read frequency, center, spread, clusters, gaps, and possible outliers.

Histograms: Read the Bars Correctly

A histogram groups numerical data into intervals. Each bar represents all observations that fall inside its interval.

50–59 60–69 70–79 80–89 90–99 100–109 Score interval Frequency
Example histogram: the \(80\text{–}89\) bar represents every observation in that score interval.
⚠
Histogram trap: A bar labeled \(80\text{–}89\) is an interval, not a single score.

Box Plots: The Five-Number Story

A box plot summarizes a distribution with five key values.

Minimum Q₁ Median Q₃ Maximum IQR = Q₃ − Q₁
The box contains the middle 50% of the data.
Q₁
25th percentile
Median
50th percentile
Q₃
75th percentile
\[ \boxed{ IQR=Q_3-Q_1 } \]
05

Relationships & Models

Scatterplots, trends, predictions, and mathematical models.

Scatterplots: Finding Relationships

Positive association
Negative association
No clear association
Critical idea: Association does not automatically mean causation.

Line of Best Fit

A line of best fit describes the overall trend in a scatterplot and can be used to make predictions.

\[ y=mx+b \]
\(m\): slope
The predicted change in \(y\) for each 1-unit increase in \(x\).
\(b\): intercept
The predicted value of \(y\) when \(x=0\).

Example: A model is \[ y=4.2x+17. \] If \(x\) increases by \(1\), the predicted value of \(y\) increases by \(4.2\).

Predictions: Observed vs. Predicted

x prediction
A model predicts a value; it does not claim that the predicted value was actually observed.
Interpolation: Prediction inside the observed \(x\)-range is generally more reliable than extrapolation far outside it.

Linear, Quadratic & Exponential Models

Linear
Constant additive change.
\(+5,\ +5,\ +5\)
Quadratic
Differences change at a constant rate.
\(+2,\ +4,\ +6\)
Exponential
Constant multiplicative change.
\(\times2,\ \times2,\ \times2\)

Common model forms include:

\[ y=mx+b \qquad\qquad y=a(b)^x \]
06

Probability & Statistical Studies

Probability, samples, margin of error, and evidence.

Probability: The Basic Framework

\[ \boxed{ P(A) = \frac{ \text{favorable outcomes} }{ \text{total outcomes} } } \]

Example: A bag contains \(5\) blue balls and \(3\) red balls. If one ball is selected at random, the probability of selecting a red ball is:

\[ P(\text{red}) = \frac{3}{8} \]
Complement: If \(P(A)\) is known, then \[ P(\text{not }A)=1-P(A). \]

Conditional Probability: “Given That” Changes the Denominator

The phrase given that tells you to restrict the group you are considering.

Plays sports Doesn’t Total
Juniors 30 20 50
Seniors 18 32 50
Total 48 52 100

Question: What is the probability that a student plays sports given that the student is a senior?

\[ P(\text{sports}\mid\text{senior}) = \frac{18}{50} = 36\% \]
Denominator alert: Once the question says “given that the student is a senior,” the relevant universe is the \(50\) seniors—not all \(100\) students.

Samples & Populations

POPULATION 10,000 students the group we want to understand RANDOM SAMPLE 500 students the group we actually study
A sample is used to learn about a larger population.
Why random? A properly selected random sample can better represent the population and reduce systematic selection bias.

Margin of Error

Suppose a poll estimates that \(54\%\) of students prefer option A, with a margin of error of \(\pm3\%\).

\[ 54\%\pm3\% \]

The corresponding interval is:

\[ 51\%\le p\le57\% \]
54% 51% 57% estimate ± margin of error
Margin of error describes an interval around an estimate.

Observational Studies vs. Experiments

Observational study
Researchers observe subjects without assigning a treatment.
Experiment
Researchers assign treatments or conditions and measure outcomes.

When subjects are randomly assigned to treatment groups, differences between groups are less likely to be explained by pre-existing characteristics. This provides stronger evidence for a causal relationship.

⚠
Remember: An association between two variables does not, by itself, prove that one variable causes the other.
07

The SAT Data Analysis Playbook

A repeatable method for unfamiliar questions.

The 5-Step Data Analysis Method

01 · READ
What exactly is the question asking?
02 · IDENTIFY
What type of problem is this?
03 · TRANSLATE
Turn words, tables, or graphs into mathematics.
04 · CALCULATE
Do only the mathematics you need.
05 · VERIFY
Does the answer make sense in context? Check the units, magnitude, and wording.

The SAT Data Analysis Trap List

❌ Wrong denominator
❌ Median without ordering the data
❌ Mean vs. median confusion
❌ Percent change vs. percentage points
❌ Treating histogram bins as individual values
❌ Correlation = causation
❌ Prediction = observed value
❌ Ignoring units
❌ Using the wrong population for “given that”
❌ Calculating what the graph already tells you

One Final Mindset Shift

\[ \boxed{ \text{Data}\neq\text{Noise} } \]

The SAT does not give you data to overwhelm you.
It gives you data to interpret.

READ → IDENTIFY → TRANSLATE → CALCULATE → VERIFY

Ready for the Data Analysis Challenge?

Put the ideas into practice with 24 original SAT-style questions across 12 high-value Data Analysis categories.

Ratios · Percentages · Mean · Median · Spread · Tables · Histograms · Box Plots · Scatterplots · Models · Probability · Sampling

Practice Data Analysis →
SATMath800.com · From SAT to University with Dr. Aytekin

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