List three ways you could organize a list of 20 numbers so a stranger could quickly understand the data.
A dot plot places one dot above a number line for every data value. It's the fastest way to see the shape, the range, and any clusters or gaps in a small-to-medium data set.
| 28 workers, ages 16–42 | Range = 42 − 16 = 26 years. The dots pile up between 16–25, then thin out — this is a right-skewed (stretched-right) distribution. |
A histogram groups data into equal-width intervals (bins) and shows the count in each bin as a bar. You lose the individual data values, but it's easier to read totals for large data sets.
| Points Scored | 40–49 | 50–59 | 60–69 | 70–79 | 80–89 | 90–99 |
|---|---|---|---|---|---|---|
| Frequency | 3 | 8 | 6 | 2 | 0 | 1 |
A distribution concentrated on one side with a "tail" stretching the other way is skewed toward the tail. A distribution with roughly equal spread on both sides of the center is symmetric. Skewed data is exactly why the mean can be misleading — an average age of 22 doesn't represent a group where 20 out of 28 workers are younger than that.
Choosing a display: dot plots show every individual value; histograms bin values into ranges (good for large sets); box plots show only the five-number summary (great for comparing groups). Shape vocabulary: symmetric, skewed left/right (the skew names the side of the long tail).
| 30 test scores from 55 to 100 — best display? | Histogram — bins tame a big spread. A dot plot with 30 dots over 45 values gets noisy. |
| A histogram's tail stretches toward low values. | Skewed left. The tail, not the pile, names the skew. |
Find the median of: 12, 15, 18, 22, 25, 30 (six values, already sorted).
The first quartile (Q₁) is the median of the lower half of the data. The third quartile (Q₃) is the median of the upper half. Together with the min, median, and max, these five numbers form the five-number summary.
| 52,60,66,66 | 68,72,72,73 | 74,75,80,82 | 84,91,92,98 | 16 values. Median = (73+74)/2 = 73.5. Q₁ = median of lower 8 = (66+68)/2 = 67. Q₃ = median of upper 8 = (82+84)/2 = 83. |
A box plot (box-and-whiskers) plots the five-number summary on a number line: whiskers stretch from the min to Q₁ and from Q₃ to the max, and a box spans Q₁ to Q₃ with a line at the median.
| min=52, Q₁=67, med=73.5, Q₃=83, max=98 | Draw whiskers 52→67 and 83→98; draw a box from 67 to 83 with a vertical line at 73.5 |
| If Q₃ = 76 on a quiz | 76 is greater than or equal to 75% of all the other scores on that quiz |
Quartiles cut ordered data into four equal-count chunks: Q2 is the median; Q1 the median of the lower half; Q3 the median of the upper half. The box in a box plot spans Q1 to Q3 (the middle 50%), the line inside is the median, whiskers reach the min and max. Always sort first.
| Data: 3, 5, 6, 8, 9, 11, 14. Find the quartiles. | Median 8; Q1 = 5 (middle of 3,5,6); Q3 = 11 (middle of 9,11,14). |
| What fraction of data sits inside the box? | About half — the box is the IQR, from the 25th to 75th percentile. |
Find the median of: 0, 1, 1, 1, 2, 2, 3, 3, 3, 3, 4, 6 (12 values, already sorted)
A measure of central tendency is a single number that represents a data set as a whole. The two most common are the mean (add everything, divide by the count) and the median (the middle value once sorted).
| 0,1,1,1,2,2,3,3,3,3,4,6 (passwords, 12 people) | mean = 29/12 ≈ 2.4. median = (2+3)/2 = 2.5 |
When data is organized in a frequency table, multiply each value by its frequency, add those products, then divide by the total number of data points (the sum of the frequencies) — not by the number of rows in the table.
| Speed (mph) | 29 | 33 | 34 | 35 | 36 | 38 | 39 | 54 |
|---|---|---|---|---|---|---|---|---|
| # Cars | 1 | 2 | 4 | 5 | 3 | 2 | 2 | 1 |
| mean = 722 ÷ 20 | = 36.1 mph. Median = average of the 10th and 11th values (both fall in the "35" interval) = 35 mph |
| 45,78,82,85,87,89,93,95 (test scores) | mean = 654/8 = 81.75. median = (85+87)/2 = 86. The 45 is an outlier — nearly 40 points from the mean. |
| Which is the better "typical student" score? | The median (86) — 6 of the 8 students actually scored above the mean of 81.75, so the mean understates how the average student did. |
A census counts every member of a population; a sample uses only a portion. Whenever you take a sample, it must be fair — it has to reasonably reflect the overall population, or your central-tendency numbers will be biased.
| Using a basketball team's heights (mean 74.6 in) to estimate the average height of ALL high school boys | Not a fair sample — basketball players tend to be taller than typical students, so this sample overestimates the true mean height. |
Mean = balance point (add and divide); median = middle of the sorted list; mode = most frequent. The mean chases outliers; the median stands firm. Skewed data or outliers → report the median; symmetric data → mean and median agree anyway.
| Data: 1, 2, 2, 3, 100. Mean vs median? | Mean = 21.6, median = 2. One wild value drags the mean far from the crowd. |
| Even-count list: 4, 6, 10, 12. Find the median. | No single middle: average the two central values → (6 + 10)/2 = 8. |
Two data sets both have a mean of 6. Can they still look completely different? Explain.
Measures of central tendency (mean, median) describe a typical value, but they say nothing about how spread out the data is. The IQR is one measure of that spread — the difference between Q₃ and Q₁.
| Set #1: 3,3,4,4,5,5,6,6,7,8,8,9,9,10,10,11,11 (mean=7) | Q₁=4.5, Q₃=9.5 → IQR = 9.5−4.5 = 5 |
| Set #2: 5,5,6,6,7,7,8,8,9,9 (mean=7, same mean!) | Q₁=6, Q₃=8 → IQR = 8−6 = 2 — much less spread, even with an identical mean |
The standard deviation (σ for a population, sₓ for a sample) is the best overall measure of variation: it tells you, on average, how far a typical data point sits from the mean. A larger standard deviation means more spread; a smaller one means the data clusters tightly. The exact calculation is complex — we rely on the calculator for it (see the Calc tab).
| Set #1 (IQR=5): σ = 2.7 | Set #2 (IQR=2): σ = 1.4 — the standard deviations confirm what the IQRs already showed: Set #1 has more spread |
| Chick weights: mean = 3.7 oz, σ = 1.7 oz | Interpretation: on average, a typical chick's weight lies 1.7 ounces away from the mean weight of 3.7 ounces |
When you have every data point of interest, that's a population, and you use the population standard deviation, σ. When you only have a portion of the data, that's a sample, and you use the sample standard deviation, s (or sₓ). The two formulas differ slightly, so their values are usually a bit different too.
| Soda A ages: sₐ = 9.1 years. Soda B ages: s_B = 4.1 years | Soda A has far greater diversity in age — a typical Soda A drinker's age is 9.1 years from the mean, versus only 4.1 years for Soda B |
| {11,11,12,13,13} vs {3,7,11,11,11,18} | {11,11,12,13,13} has almost no spread at all — its standard deviation is closest to zero of the two. |
Center says where the data lives; spread says how tightly. Range is quick but fragile (uses only two values). Standard deviation is the workhorse: a typical distance from the mean. Larger SD = more scattered. Two classes can share a mean of 80 and be wildly different — spread is the difference.
| Class A: 78, 80, 82. Class B: 60, 80, 100. Compare. | Same mean 80. A's SD ≈ 1.6; B's ≈ 16.3 — B is far less consistent. |
| Add 5 to every data point — what happens? | Mean rises by 5; SD unchanged — the whole set slides without spreading. |
Find the IQR of the data set: 8, 17, 20, 22, 22, 23, 23, 25, 25, 25, 26, 26, 28, 28, 29, 30 (Q₁=22, Q₃=27)
1) Calculate the IQR. 2) Lower bound = Q₁ − 1.5×IQR. 3) Upper bound = Q₃ + 1.5×IQR. 4) Any data point falling outside [lower bound, upper bound] is an outlier.
| Q₁=22, Q₃=27, IQR=5 | 1.5×IQR = 7.5. Lower bound = 22−7.5 = 14.5. Upper bound = 27+7.5 = 34.5. The value 8 falls below 14.5, so 8 is the only outlier. |
| Full set: mean=23.6, sₓ=5.2 | After removing the low outlier (8): mean=24.6, sₓ=3.4 — mean rose, standard deviation shrank, exactly as expected |
On a box plot, a whisker that's unusually long, or a five-number summary where min or max sit far outside Q₁ − 1.5(IQR) or Q₃ + 1.5(IQR), signals a likely outlier. Sometimes a five-number summary alone can only guarantee a minimum number of outliers — there could always be more hidden inside the data you can't see.
| min=3, Q₁=11, med=14, Q₃=15, max=23 | IQR=4, bounds = [5, 21]. Both min (3) and max (23) fall outside — so there are at least two outliers, possibly more. |
A major outlier lies more than 3 times the IQR beyond Q₁ or Q₃ — a much stricter test than the standard 1.5×IQR outlier test. Every major outlier is automatically also a regular outlier, but not the reverse.
The 1.5×IQR rule builds fences: Q1 − 1.5·IQR and Q3 + 1.5·IQR. Anything beyond a fence is an outlier. When you find one, don't auto-delete — ask if it's a typo or a real extreme value; then report the median/IQR, which outliers can't bully.
| Q1 = 20, Q3 = 32. Is 52 an outlier? | IQR = 12; upper fence = 32 + 18 = 50. Yes — 52 > 50. |
| Same data: is 3 an outlier? | Lower fence = 20 − 18 = 2. No — 3 is inside, barely. |
Write 6/16 as a decimal rounded to the nearest hundredth.
So far we've studied quantitative data (numbers like weight or speed). Categorical data instead sorts people into categories. When we cross two categories at once — like hair color AND eye color — we summarize it in a two-way frequency table.
| Eye \ Hair | Black | Blond | Red | Total |
|---|---|---|---|---|
| Blue | 3 | 4 | 1 | 8 |
| Brown | 5 | 2 | 0 | 7 |
| Green | 1 | 1 | 3 | 5 |
| Total | 9 | 7 | 4 | 20 |
| Joint: blond hair AND blue eyes | 4/20 = 0.2, or 20% |
| Marginal: blond hair (any eye color) | 7/20 = 0.35, or 35% |
A conditional relative frequency restricts to ONE row or column first, then divides by that row or column's total — not the grand total. The word "given," or "if you have ___," is the signal to restrict first.
| P(green eyes | red hair) | Restrict to the 4 people with red hair; 3 of them have green eyes → 3/4 = 0.75 |
| P(green eyes | black hair) | Restrict to the 9 people with black hair; 1 has green eyes → 1/9 ≈ 0.11 |
| 29 go to college, 13 of them female → P(female | going to college) | 13/29 ≈ 0.45 |
| 22 total females, 13 go to college → P(going to college | female) | 13/22 ≈ 0.59 — a different answer, because the denominator (the "given" group) is different |
Two-way tables sort people by two questions at once. The denominator is everything: 'What fraction of seniors drive?' divides by the senior row total, while 'what fraction of the school are driving seniors?' divides by the grand total. Read the phrase 'of ___' to find your denominator.
| 40 of 60 athletes lift weights; 10 of 50 non-athletes do. What % of lifters are athletes? | Lifters total 50; athletes among them 40 → 80%. |
| Same data: what % of athletes lift? | 40/60 ≈ 67% — different denominator, different question, different answer. |
If two variables increase together, is that a positive or negative relationship?
Bivariate data involves two related quantitative variables, collected together and visualized with a scatter plot. A line of best fit is a straight line drawn by eye through the "center" of the data, used to estimate the trend.
| Low vs. high temperature, 10 April days | Points trend upward left-to-right — as the low temperature rises, so does the high temperature |
| Points on the line: (32,56) and (40,66) | m = (66−56)/(40−32) = 10/8 = 1.25 |
| Using the line to predict at x=42°F (low temp) | Reading up from x=42 to the line gives a predicted high temperature just above 68°F |
| Positive correlation | Both variables move in the SAME direction — as one increases, so does the other |
| Negative correlation | The variables move in OPPOSITE directions — as one increases, the other decreases |
| Person's height vs. shoe size — strong positive correlation | No causal relationship — a taller person tends to have bigger feet, but height doesn't "cause" shoe size directly |
| Number of firefighters at a fire vs. dollar damage done — strong positive correlation | No causal relationship — the lurking variable is the SIZE of the fire itself: a bigger fire causes both more firefighters to respond AND more damage |
| Ice-cream cones sold vs. people swimming — strong positive correlation | No causal relationship — the lurking variable is the outdoor temperature: hot weather causes both more ice cream sales and more swimming |
| Hours spent studying vs. GPA — strong positive correlation | This one IS causal — more hours spent studying directly causes a higher GPA |
| y = 12x + 33,766 (car cost → house value), predict at x=19,500 | y = 12(19,500)+33,766 = $267,766 — but the actual table value was $255,000, so this prediction is an overestimate |
Bivariate = two variables per person (height and shoe size), plotted as a scatter plot. Describe three things: direction (positive/negative), form (linear/curved), strength (tight/loose). And the golden warning: correlation is not causation — ice cream sales and drownings rise together because of summer, not each other.
| As altitude rises, temperature falls, tightly and straight. | Strong, negative, linear association. |
| Shoe size vs reading level in children correlate. Cause? | No — age drives both. A lurking variable creates the link. |
In y = ax + b, what does a represent?
Instead of drawing a line of best fit by eye, the calculator can compute the actual equation of the line of best fit — called linear regression — directly from the data. This is far more precise than eyeballing a line.
| Low Temp, x | 26 | 28 | 30 | 32 | 34 | 35 | 37 | 38 | 41 | 45 |
|---|---|---|---|---|---|---|---|---|---|---|
| High Temp, y | 49 | 50 | 57 | 54 | 60 | 58 | 64 | 66 | 63 | 72 |
| Calculator regression: y = 1.16x + 19 | Compare to the hand-drawn estimate from Lesson 6 (m ≈ 1.25) — close, but the calculator's value is more precise since it uses every data point, not just two. |
| y = 1.16x + 19 (low temp → high temp) | Slope: for every 1° increase in the low temperature, the model predicts a 1.16° increase in the high temperature. y-intercept: when the low temperature is 0°F, the model predicts a high of 19°F. |
The sign of the slope alone tells you whether the correlation is positive or negative — you don't need to look at a graph at all.
| y = −8.1x + 68.1 (car weight → fuel efficiency) | Negative slope → negative correlation: as weight increases, fuel efficiency (mpg) decreases |
| Predict mpg at x = 4.3 (4,300 lbs) | y = −8.1(4.3)+68.1 = 33.27 ≈ 33 mpg |
| Solve for x when y = 40 mpg | 40 = −8.1x+68.1 → x ≈ 3.47 thousand lbs ≈ 3,500 pounds |
Even a strong regression line can "miss" one or two data points badly. Graphing the scatter plot alongside the line of best fit lets you visually spot which points sit far from the trend.
| y = −0.005x + 65 (elevation → mean temperature) | The data point (5625 ft, 48°F) sits noticeably off the line — this point is an outlier relative to the regression model, even though the overall fit is otherwise strong |
The regression line is the straight line that best threads the cloud, minimizing total miss. Its slope speaks in units: 'each extra hour of study predicts +6.5 points.' Predicting inside your data range is interpolation (trustworthy); beyond it is extrapolation (increasingly fiction).
| ŷ = 2.3x + 10, x = grams of fertilizer, y = cm growth. Interpret 2.3. | Each extra gram predicts 2.3 cm more growth, on average. |
| Data covers x = 1 to 9. Predict at x = 25? | Extrapolation — the linear pattern may not survive that far out; treat with suspicion. |
Does a parabola open upward look more like a "U" or an "n"?
Not every scatter plot is best fit by a straight line. Exponential curves grow (or decay) increasingly fast in one direction, always curving the same way. Quadratic curves have a single turning point — they decrease then increase, or increase then decrease, forming a U or an upside-down U.
| Data steadily rises, curving upward more and more steeply | Exponential — the "runaway growth" shape, no turning point |
| Data rises to a peak, then falls back down | Quadratic — has exactly one turning point (a maximum or minimum) |
| Data falls in a fairly straight diagonal line | Linear — no curvature at all |
The calculator can also fit an exponential model, y = a(b)ˣ, the same way it fits a linear model — just choose Exponential Regression instead of Linear Regression.
| Day, x | 0 | 1 | 3 | 4 | 6 | 7 |
|---|---|---|---|---|---|---|
| Flu Cases, y | 16 | 18 | 22 | 25 | 33 | 35 |
| Exponential regression: y = 15.92(1.12)ˣ | Since b=1.12 > 1, this models 12% daily growth — every additional day multiplies the case count by 1.12 |
| Predict at x=14 (two weeks) | y = 15.92(1.12)¹⁴ ≈ 78 cases |
| Cost per widget vs. number produced: drops, then rises again | A quadratic model fits — the U-shape (dropping then rising) can't be captured by a straight line or a one-directional exponential curve |
Linear and exponential models that fit similarly well NEAR the data can diverge wildly when extrapolated FAR beyond it — exponential models always eventually grow faster than linear ones.
| Linear: y=10.83x+114.07 vs. Exponential: y=121.09(1.06)ˣ at x=10 | Linear ≈ 222, Exponential ≈ 217 — very close |
| Same two models at x=30 (far extrapolation) | Linear ≈ 439, Exponential ≈ 695 — no longer close at all; the exponential model predicts much faster growth over time |
Not every cloud is a line — the calculator also fits exponential and quadratic models. Choose by the data's shape: constant differences → linear; constant ratios / multiplicative growth → exponential; rises then falls (or has a clear turning point) → quadratic.
| Population doubling every decade — which model? | Exponential: equal ratios between evenly spaced readings are the tell. |
| Projectile height data over time — which model? | Quadratic: up, turn, down is the parabola's signature. |
If a correlation coefficient r is close to 0, is the linear fit strong or weak?
The correlation coefficient, r, is a single number between −1 and 1 that measures exactly how well a linear model fits bivariate data. The closer |r| is to 1, the better the fit; the closer to 0, the worse.
| r = 0.999 | Very good fit — points fall almost exactly on a line |
| r = 0.930 | O.K. fit — a decent, but not perfect, linear trend |
| r = 0.776 | Poor fit, very scattered |
| r = 0.133 | Almost no fit at all |
| r = −0.945 | Negative correlation (negative sign) that is a very good, strong fit (close to −1) |
You can often estimate roughly what r should be just by how tightly the points hug an imaginary line, and in which direction that line points.
| Scatter plot: strong upward trend, points close to a line | Expect r to be positive and close to 1 (e.g. r=0.88, not r=0.28 or r=1 exactly, unless the fit is perfect) |
| Scatter plot: downward trend, points tightly clustered | Expect r close to −1 — the tighter the cluster around a negative-sloped line, the closer to −1 |
| Brent Crude: r=0.973 vs. WTI Crude: r=0.924 (both predicting gas price) | Choose the Brent Crude model — its r-value is closer to 1, meaning it's the more reliable predictor |
The correlation coefficient r (between −1 and 1) scores a linear fit: the sign gives direction, the absolute value gives strength. Near ±1 = tight line; near 0 = no linear pattern (a perfect curve can still score near 0). r has no units and doesn't care which variable is x.
| Rank strength: r = −0.92, r = 0.35, r = 0.78 | −0.92 strongest (closest to ±1), then 0.78, then 0.35. Sign is direction, not strength. |
| r = 0.05 for clearly curved data | Not 'no relationship' — just no linear one. r only measures straightness. |
If a model predicts y=50 but the actual observed value is y=42, what is the residual?
A residual is the difference between the observed y-value and the predicted y-value: residual = observed − predicted. The r-value tells you how well a model predicts overall, but residuals reveal whether the model TYPE (linear, exponential, quadratic) was the right choice in the first place.
| Model: y=1.3x+73.7. Data point: (11,94) | Predicted: y=1.3(11)+73.7=88. Residual = 94 − 88 = 6 |
| Skydiver speed vs. time: r = 0.94 (a strong-looking fit) | But the residual plot shows a clear rise-then-fall pattern — this means the LINEAR model is not actually appropriate, despite the strong r-value; an exponential model would fit better |
| Study hours vs. GPA: r=0.88, residuals randomly scattered | This model is MORE appropriate than the skydiver model above, even though its r-value (0.88) is lower than the skydiver's (0.94) — because its residuals show no pattern. |
| Residuals randomly scattered above/below the axis | Linear model is appropriate |
| Residuals show a clear curve (e.g. dip then rise) | Linear model is NOT appropriate — try exponential or quadratic instead |
A residual = actual − predicted: positive means the point beat the line's prediction, negative means it fell short. The residual plot is the fit's report card: random scatter around zero → the model suits the data; a curve or fan pattern → a straight line was the wrong tool.
| Actual score 88; line predicted 82. Residual? | 88 − 82 = +6 — the student outperformed the model by 6 points. |
| Residual plot shows a clear U-shape. | The line missed a curved pattern — try a quadratic model instead. |
By hand: find the median of 4, 8, 6, 10, 6. (Sort first!)
This is the calculator authorized for this course. These steps let you check any five-number summary, standard deviation, or regression equation from this unit numerically, rather than by hand.
Calculator stats workflow: enter data into two named lists, then run one-variable stats (mean, SD, five-number summary) or a regression (a, b, and r) from the statistics menu. Most errors are list errors — mismatched lengths or a stray deleted cell — so eyeball the lists before trusting the output.
| Regression output: a = 58.1, b = 6.5, r = 0.997. | For the a + bx form: ŷ = 58.1 + 6.5x, an excellent fit (r near 1). |
| Error: 'dimension mismatch.' | The two lists have different lengths — every x needs exactly one partner y. |
These problems pull from earlier units on purpose. Switching between skills feels harder in the moment, but it helps you remember longer and matches how a real exam mixes topics. Try each one on paper first, then reveal.Estos problemas provienen de unidades anteriores a propósito. Cambiar de una destreza a otra se siente más difícil en el momento, pero te ayuda a recordar por más tiempo y se parece a cómo un examen real mezcla los temas. Intenta cada uno en papel primero, luego revela la respuesta.