9.2 Introduction to Statistics, Part 2

Part 1 gave you location and spread. Part 2 adds pictures, a rule for outliers, and the linear relationship between two numerical series. Statistical Analysis still sits in Advanced Techniques (26% of CMT Level I). Independent OpenExamPrep material for this unit is the scatterplot-and-regression toolkit technicians use when they stop looking at one series in isolation.

The value of data visualization

A table of 60 daily returns can hide a bimodal split (quiet days plus crash days), a trend in the scatter, or a single typo-sized print. Data visualization is how you see shape, outliers, clusters, and relationships before you trust a mean or a correlation.

For one series, technicians already live in pictures: histograms, equity curves, box plots, and the price chart itself. For two series, the scatterplot is the native picture. The numerical summary (Pearson r, a regression slope) is a compression of that picture. Compression throws information away. Four different clouds can share a similar r and still tell different stories: a tight line, a curve, a cloud with one outlier manufacturing the r, or two blobs. Look at the scatter before you quote r. That is the value of visualization on this exam.

Visualization also catches bad data. A decimal in the wrong place becomes a point in the far corner. If you only compute the mean, the error is baked in. If you plot, you see it.

Calculating outliers

An outlier is an observation that sits far from the bulk of the sample—far enough that you should not treat it as just another point without comment. Outliers can be errors, regime breaks, or real tail events. Charts are full of the third kind. The skill is to flag them with a stated rule.

Two Level I-useful rules:

1. Tukey fences (IQR rule). Sort the sample. Find Q1 (25th percentile) and Q3 (75th percentile). IQR = Q3 − Q1.

• Lower fence = Q1 − 1.5 × IQR
• Upper fence = Q3 + 1.5 × IQR

Any point below the lower fence or above the upper fence is flagged as an outlier under this rule. Some texts use 3 × IQR for extreme outliers. Unless the stem says otherwise, 1.5 × IQR is the standard fence.

2. Z-score rule. Compute z = (x − mean) / SD. Flag |z| > 3 (sometimes 2.5 or 2; the stem will say). This rule is pulled around by the outlier itself: a crash inflates SD and can hide the next-worst day. Tukey fences use quartiles, so they resist that hiding.

Worked Tukey example. Eight weekly percentage returns: −1, 0, 1, 1, 2, 2, 3, 12.

n = 8. The quartile positions using (n+1)/4:

• Q1 sits at position 2.25: between 0 and 1 → Q1 = 0.25
• Q3 sits at position 6.75: between 2 and 3 → Q3 = 2.75
• IQR = 2.50
• Lower fence = 0.25 − 1.5 × 2.50 = −3.50
• Upper fence = 2.75 + 1.5 × 2.50 = 6.50

12 sits above 6.50, so it is an outlier. −1 is inside the lower fence, so it is not. After you flag 12, ask why: a one-week panic, a bad tick, a split not adjusted? Do not silently drop it from a risk sample just to make the SD pretty. Do not silently leave a data error in a correlation.

Scatterplots and the three features they describe

A scatterplot graphs pairs (x, y) as points. Each point is one observation of two numerical variables: this week's index return versus this week's stock return; today's range versus today's volume; an indicator reading versus the next week's return.

Read three features, in this order:

1. Direction. As x increases, does y tend to increase (positive), decrease (negative), or show no monotone tendency (none)?
2. Form (linearity). Is the cloud well described by a straight line, a curve, or no simple form (a blob, clusters, a fan)?
3. Strength. How tightly do the points hug that form? A tight cigar is strong. A wide shotgun blast around a faint slope is weak. Strength is about scatter around the form, not about whether the slope is steep. A steep line with huge residuals is a strong-looking slope and a weak relationship.

Exam trap: calling a steep but noisy cloud "strong" because the line is tilted. Steep is slope. Tight is strength.

A fan that opens to the right is spread that changes with x—common when you plot raw price against time, which is one reason technicians often work in returns. You do not need extra vocabulary on every stem. You do need to see that a single straight line is a poorer summary when the cloud's width is changing.

Pearson's r

Pearson's product-moment correlation coefficient r measures the strength and direction of a linear association between two numerical variables. Conceptually,

r = covariance(x, y) / (sₓ × sᵧ).

It is unit-free. It ranges from −1 to +1.

• r = +1: all points sit on an upward straight line.
• r = −1: all points sit on a downward straight line.
• r = 0: no linear association. A U-shaped curve can have r near 0 and still be a perfect nonlinear pattern. That is why visualization comes first.

Those bands are teaching aids, not laws. With n = 8, an r of 0.6 can be luck. With n = 800, an r of 0.2 can be a real but small linear link. r does not care about sample size in its formula; your confidence in r does.

r is also symmetric: corr(x, y) = corr(y, x). It is not a slope. Doubling every y leaves r unchanged if x is unchanged; the regression slope would double. Mixing r with slope is a frequent wrong answer.

r is computed on numerical pairs. Do not quote Pearson r between candle color and a sector name.

Correlation versus causation

A large |r| means the two series move together linearly. It does not mean x causes y. Three other stories are always on the table:

• Reverse causation. y is driving x (or traders are using y to set x).
• Confounding. A third factor—rates, liquidity, the index—moves both.
• Coincidence / data mining. You searched 80 indicators and kept the one with the prettiest r on this sample.

Market example: weekly returns of two semiconductor stocks can show a high positive r because both are beta to the same sector and to the same risk-on tape, not because Stock A's print causes Stock B's print. Another: an oscillator can correlate with past returns because it is built from them. That r is almost a tautology; it is not evidence that the oscillator causes the next move.

For inference you still need a story, later-sample tests, and humility. On Level I, the tested sentence is: correlation is not causation.

What a linear model determines

A linear model for two variables says that y is a straight-line function of x, plus leftover error:

y = a + b x + error.

Ordinary least squares (OLS) chooses a (intercept) and b (slope) to minimize the sum of squared residuals (the vertical misses). The fitted line determines:

• A slope: the predicted change in y for a one-unit increase in x.
• An intercept: the predicted y when x = 0 (which may or may not be a meaningful x).
• A fitted value ŷ for each observed x, and therefore a residual e = y − ŷ.

The linear model does not determine causation, uniqueness (another x might fit as well), or that the true relationship is linear. It determines the best-fitting straight line in the least-squares sense for this sample.

The linear regression equation

Write the prediction as ŷ = a + b x.

The slope and intercept connect to r:

• b = r × (sᵧ / sₓ)
• a = ȳ − b x̄

Worked numbers. Suppose r = 0.80, sₓ = 2, sᵧ = 5, x̄ = 10, ȳ = 50.

• b = 0.80 × (5 / 2) = 2.0
• a = 50 − 2.0 × 10 = 30
• ŷ = 30 + 2x

If x = 12, ŷ = 30 + 24 = 54. If the actual y is 58, the residual is +4.

Read the slope in words: a one-unit increase in x is associated with a +2 unit change in predicted y in this sample. If x is the index return in percent and y is the stock return in percent, b is a sample beta against that index over that window.

Regression in technical studies

Technicians use the same equation in several disguises:

• Price versus time. A linear trendline fitted through closes is a regression of price on time. The slope is average points (or dollars) per bar. On a log y-axis the analogous fit is about constant proportional growth; that choice belongs with chart scales, but the algebra is still a + b x.
• Instrument versus benchmark. Regress the stock's returns on the index's returns. The slope is beta; the intercept is a sample leftover often called alpha. Relative-strength work later in Level I is a cousin: it asks who is stronger, not only how tightly they co-move.
• Indicator versus later return. Scatter this week's reading of an indicator against next week's return, then fit a line. A steep in-sample line is a research clue, not a live edge, until it survives a later sample.
• Pairs / spread. Regress one name on another. The residual is a spread. Mean-reversion traders watch that residual; they are betting the linear relationship holds well enough to fade extremes. If r collapses, the spread model is the risk.

In all of these, regression is a description of past co-movement plus a machine for fitted values. It is not a promise that the slope will persist. The quantitative-methods unit later will add backtests and bias. Here, get the equation and the meaning of the two coefficients right.

Coefficient of correlation versus coefficient of determination

Keep the names straight.

r is the coefficient of correlation (Pearson). It is signed. It answers: how strong and in which direction is the linear association?

R² (R-squared) is the coefficient of determination. In simple linear regression (one x), R² = r². It lies between 0 and 1 (or 0% and 100%). It answers: what fraction of the variance of y does the fitted line explain?

From the worked example, r = 0.80, so R² = 0.64. The line explains 64% of the sample variance of y. 36% is leftover residual variance. R² does not keep the sign: r = −0.80 also gives R² = 0.64. Direction lives in r (or in the slope b), not in R².

Exam trap: "r = 0.70, so the model explains 70% of variance." No. It explains 49%. Another trap: "R² = 0.49 means a 49% chance the relationship is real." No. R² is not a probability that the model is true.

On the exam

A typical stem gives you a scatter description or an r. Walk it: direction, form, strength first; then r if the form is linear; then refuse causation; then, if they ask about explained variance, square r. If they give r, sₓ, sᵧ, and the two means, you can write the regression line. Independent practice on those steps beats memorizing software output labels.

Key Takeaways

• Visualize before you trust a mean, an r, or a slope
• Tukey: fences at Q1 − 1.5×IQR and Q3 + 1.5×IQR
• Scatterplots: direction, form/linearity, strength
• r is signed linear association; R² = r² is explained variance in simple regression
• Linear model: ŷ = a + bx, least squares, not causation