Center and spread — mean, median, IQR, standard deviation.
The dataset (9 test scores, sorted)
12,15,18,20,22,25,30,31,35
Step01of 06
Two questions describe almost any dataset: where is the center? and how spread out is it? Each has two common measurements.
Mean (average)
→sum ÷ count — sensitive to outliers
Median (middle value)
→middle of sorted data — resistant to outliers
IQR (interquartile range)
→Q3 − Q1, the spread of the middle 50%
Standard deviation
→typical distance from the mean
A1.L.1 — Center and spread — mean, median, IQR, standard deviation.
The dataset (9 test scores, sorted)
12,15,18,20,22,25,30,31,35
Step 01 of 06
Two questions describe almost any dataset: where is the center? and how spread out is it? Each has two common measurements.
Mean (average)
→sum ÷ count — sensitive to outliers
Median (middle value)
→middle of sorted data — resistant to outliers
IQR (interquartile range)
→Q3 − Q1, the spread of the middle 50%
Standard deviation
→typical distance from the mean
Step 02 of 06
Mean. Sum the values, divide by count.
xˉ=912+15+18+20+22+25+30+31+35=9208≈23.1
The bar over x ("x-bar") is the standard symbol for sample mean.
Step 03 of 06
Median. Sort, then pick the middle value. With 9 numbers the middle is the 5th:
12,15,18,20,22,25,30,31,35
median=22
For an EVEN count, average the two middle values. The median doesn't care how extreme the outliers are — it only cares about position.
Step 04 of 06
IQR — interquartile range. Find the median of the LOWER half and the median of the UPPER half (excluding the overall median).
lower half: 12,15,18,20⇒Q1=215+18=16.5
upper half: 25,30,31,35⇒Q3=230+31=30.5
IQR=Q3−Q1=30.5−16.5=14
The middle 50% of the data spans 14 units. IQR is the spread measure that ignores the tails.
Step 05 of 06
Standard deviation is the typical distance from the mean. The full formula:
s=n−1∑(xi−xˉ)2
You'll usually use a calculator or spreadsheet for SD — the value here is s≈7.84. The intuition: about 68% of values land within ±1 SD of the mean, ~95% within ±2 SD.
Step 06 of 06
Which to pick? The right measure depends on the data shape and the presence of outliers.
Symmetric data, no outliers
→mean + standard deviation
Skewed data or outliers
→median + IQR
Income, home prices
→median (a few billionaires would distort the mean)
Test scores, heights
→mean + SD usually fine
Key insight
Mean and SD describe symmetric, well-behaved data. Median and IQR are the outlier-resistant pair. The choice isn't about preference — it's about what the data looks like.
A1.L.2 — Box plots and histograms.
Same dataset
12,15,18,20,22,25,30,31,35
Step 01 of 05
Five-number summary is the foundation for a box plot. Min, Q1, median, Q3, max.
Min
→12
Q1
→16.5
Median
→22
Q3
→30.5
Max
→35
Step 02 of 05
Build the box plot. Draw a box from Q1 to Q3, a vertical line at the median, and "whiskers" extending from the box to the min and max.
The box covers the middle 50% (IQR). Whiskers reach the extremes.
The shape tells you a lot. A median line CLOSE to one end means the data is skewed in the opposite direction. Long whiskers mean wide spread in the tails.
Step 03 of 05
Histograms bin the data into intervals (usually equal-width) and count how many values fall in each bin. Bin choice is part of the design — too few bins hides shape, too many makes noise.
For our 9 values, bins of width 5 starting at 10 give:
Bins of width 5. Each bar's height = count of values in that bin.
Step 04 of 05
What histograms tell you that box plots don't.
Number of peaks (modes)
→visible in histogram, hidden in box plot
Bin-by-bin distribution
→histogram shows it, box plot summarizes
Symmetry / skew
→both show it
Outliers
→box plot flags them more cleanly
Step 05 of 05
When each shines.
Comparing 2+ groups side by side
→box plots stacked vertically
Showing distribution shape
→histogram
Spotting outliers fast
→box plot
Big datasets
→either works; histogram more informative
Small datasets (n < 20)
→box plot more readable
Key insight
Box plot = compact summary of position and spread, perfect for comparing groups. Histogram = full shape of the distribution. Same data, different questions answered.
A1.L.3 — Two-way frequency tables.
The survey
100 students were asked: do you own a smartphone, and do you play a competitive sport? Results are tallied below.
Step 01 of 05
A two-way frequency table cross-tabulates two categorical variables. Each cell counts how many subjects fit BOTH categories.
Plays sport
No sport
Total
Smartphone
32
48
80
No phone
8
12
20
Total
40
60
100
The corner cell (100) is the grand total — every student counted exactly once.
Step 02 of 05
Three flavors of "fraction" come from this table, and they answer different questions. Get the vocabulary right.
Joint
→"What % play sports AND own a phone?" → cell ÷ grand total = 32/100=32%
Marginal
→"What % play sports overall?" → row or column total ÷ grand total = 40/100=40%
Conditional
→"What % of phone owners play sports?" → cell ÷ row total = 32/80=40%
Same percentage. Owning a phone doesn't change the rate of playing sports — the two variables are independent in this dataset.
Step 04 of 05
Contrast — a table where they ARE related. Suppose instead the cells were 64 / 16 (sport / no-sport) for phone owners, and 16 / 4 for non-owners. Then:
P(sport∣phone)=8064=80%
P(sport∣no phone)=2016=80%
Still 80% in both rows — also independent (just at a different rate). For dependence, the two conditional rates have to DIFFER.
Step 05 of 05
Reading the question carefully matters more than the math. Confusing joint with conditional is the most common mistake.
"and"
→joint — divide by grand total
"of all students"
→marginal — divide by grand total
"given that" / "of"
→conditional — divide by the SUBSET total
Key insight
Three numbers pop out of every two-way table cell: joint (cell / grand), marginal (margin / grand), conditional (cell / margin). Match the wording of the question to the right denominator.
A1.L.4 — Linear regression and correlation.
The data — hours studied vs. test score (10 students)
When two numerical variables vary together, a scatter plot displays each pair as a single dot. The cloud's shape tells you whether a relationship exists.
Dots trend up to the right
→positive association
Dots trend down to the right
→negative association
Dots scatter randomly
→no association
Dots form a curve
→nonlinear association (regression line is wrong tool)
Step 02 of 05
Line of best fit. When the cloud is roughly linear, fit a straight line that minimizes the total squared distance from the dots to the line. Calculator output for our data:
y^=4.0x+52.4
The hat over y ("y-hat") signals a PREDICTED value. This isn't the actual data — it's the line's estimate of y for any given x.
Step 03 of 05
Visualize the data with the regression line. Each dot is one student; the line is the best linear summary.
x = hours studied, y = test score. Line: y^=4.0x+52.4.
Step 04 of 05
Interpret slope and intercept in context.
Slope =4.0
→every extra hour of study predicts about 4 more points
Intercept =52.4
→predicted score with 0 hours studied (often a meaningful "baseline"; sometimes nonsense — check the context)
Predict at x=6.5
→y^=4.0(6.5)+52.4=78.4 points
Predict only WITHIN the range of your data (x∈[1,10]). Extrapolating to x=100 would predict 452 — nonsense, since tests cap at 100.
Step 05 of 05
Correlation coefficient r measures how tightly the dots hug the line.
r=+1
→perfect positive linear (every dot ON the line)
r≈+0.9
→strong positive (our dataset)
r=0
→no linear relationship
r≈−0.9
→strong negative
r=−1
→perfect negative
Correlation does NOT mean causation. Strong r says the variables move together; it doesn't say one CAUSES the other. Ice cream sales and shark attacks are both correlated with summer — neither causes the other.
Key insight
Scatter plot shows the relationship. Regression line summarizes it. Slope and intercept give you a prediction rule. Correlation r says how trustworthy that line is. Causation needs more evidence than any of these can provide.
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