9.1 Introduction to Statistics, Part 1

Statistical Analysis sits inside Advanced Techniques, which carries 26% of CMT Level I—the 132-question, 2-hour sitting (120 scored plus 12 unscored pilots). Charts already summarize price. This unit asks you to summarize samples of numbers with the same honesty: what is a typical value, how spread out are the values, and how unusual is today's print relative to that history. Independent OpenExamPrep teaching for these CMT Level I statistics topics stays on definitions, a few worked numbers, and the traps that show up in multiple-choice stems. Independent CMT Level I practice by OpenExamPrep is at /practice/cmt.

What statistics is

Statistics is the discipline of collecting, organizing, summarizing, and analyzing data, then drawing conclusions under uncertainty. For a technician the data are usually prices, returns, volume, breadth, indicator values, and related series. A candlestick already compresses four prices. A moving average compresses many closes. This chapter names the summary tools so you can say what a mean, a standard deviation, or a z-score is doing—and what it is not doing.

Two jobs sit under that umbrella.

Descriptive versus inferential statistics

Descriptive statistics summarize the data set you actually have. The 20-day mean return, the median daily range for this quarter, a histogram of those ranges, and the sample standard deviation of this lookback are descriptive. They do not claim anything about next year. They describe this sample.

Inferential statistics use a sample to make a statement about a population (or about a data-generating process) that you did not fully observe. Examples: treating 252 daily returns as evidence about the typical volatility of this instrument; asking whether a mean return is distinguishable from zero; converting a z-score into an approximate probability. Inference always carries uncertainty. The sample could be unrepresentative. The process could have changed.

Exam trap: calling a 14-day RSI reading "the population RSI of the stock." It is a statistic computed on a sample window.

Sample versus population

A population is the complete set of observations you care about. In markets that set is often unavailable or open-ended: every future daily return of an index, or every tick that will ever print. A sample is the subset you actually measure—the last 60 sessions, this contract's life, this earnings-to-earnings window.

A parameter describes a population (often Greek: μ for a population mean, σ for a population standard deviation). A statistic describes a sample (x̄ for a sample mean, s for a sample standard deviation). Technicians almost always compute statistics. Calling a 20-day average μ is sloppy; it is x̄ for that window.

Sampling choices are the hidden assumption on every chart. A 20-bar sample during a panic is not the same population as a 20-bar sample during a quiet range. If the process is not stable, you do not have draws from one population. That warning returns in the probability section as a failure of identically distributed observations.

Two main types of data

Level I needs a consistent split. Use qualitative (categorical) versus quantitative (numerical) as the two main types.

Qualitative / categorical data label kind, not amount. Values are names or categories. Nominal categories have no order (sector: energy versus utilities; instrument: future versus stock). Ordinal categories have order but not a trustworthy arithmetic difference (sentiment bucketed as bearish / neutral / bullish). You can count frequencies and name a mode. You cannot honestly add "utilities + 2."

Quantitative / numerical data are numbers that measure how much or how many. Discrete numerical data are counts (number of advancing issues, number of new highs, tick count). Continuous numerical data can in principle take any value in an interval (price, return, ATR). Prices print on a tick grid, but Level I still treats price and return as numerical measurements you can average.

Keep the pair qualitative versus quantitative (equivalently categorical versus numerical) as the two main types. Discrete versus continuous is a split inside quantitative data, not a third main type.

Exam trap: treating a candle color (green/red) as a number you can average. Color is categorical. The close is numerical.

Three common measures of central tendency

A measure of central tendency is a single number that locates the center of a quantitative sample (or the most common category for a qualitative sample).

The arithmetic mean (ordinary mean) is the sum of the observations divided by the count. For a sample, x̄ = (Σ xᵢ) / n. It uses every value, so it is efficient when the sample is well behaved—and sensitive to outliers. One crash day pulls the mean of a short window hard.

The median is the middle value after you sort the sample (or the average of the two middle values if n is even). Half the observations are at or below it, half at or above. It resists a single wild print. For a skewed return sample, the median is often the more honest "typical day."

The mode is the most frequent value. For categorical data it is often the only sensible center (the most common sector in a watchlist). For continuous prices a raw mode may not exist unless you bin the data. A session histogram's tallest bin is a modal price—the price that traded most often.

Tiny numeric example

Five daily closes: 10, 12, 12, 13, 23.

• Mean = (10 + 12 + 12 + 13 + 23) / 5 = 14
• Median = 12 (the third value in the ordered list)
• Mode = 12 (it appears twice; every other value appears once)

The 23 is an outlier relative to the cluster at 10–13. It lifted the mean to 14, above the median and the mode. If a stem asks which measure is most affected by the 23, the answer is the mean.

Alternative means and why technicians use them

The arithmetic mean is not the only mean.

The trimmed mean drops a stated fraction of the lowest and highest observations, then takes the arithmetic mean of what remains. On the five closes above, drop the single lowest and single highest (10 and 23) and the remaining values are 12, 12, 13. Trimmed mean = 37 / 3 ≈ 12.33, near the median. Trimmed means are robust location tools: they still use more than one central point, but they refuse to let a crash bar dominate a short window.

The harmonic mean of positive numbers is n / Σ(1/xᵢ). It is the right average when you are averaging rates and the quantity in the denominator is what you hold fixed. Classic finance example: the average P/E of two stocks when you invest equal dollars in each is the harmonic mean of the two P/Es, not the arithmetic mean. P/Es of 10 and 30 have arithmetic mean 20 and harmonic mean 15. Technician-adjacent example: if you buy a fixed dollar amount of a stock at several different prices (dollar-cost averaging), the average purchase price is the harmonic mean of those prices. Harmonic means of numbers that can be zero or negative are not defined in the usual formula—do not harmonic-mean raw signed returns.

Why the geometric mean matters to investors

The geometric mean return is the constant per-period rate that produces the same ending wealth as the sequence of actual returns. For returns r₁ … rₙ,

Geometric mean = [(1+r₁)(1+r₂)…(1+rₙ)]^(1/n) − 1.

Investors live in wealth space. A quoted "average return" that uses the arithmetic mean answers a different question: what was the typical one-period outcome? When returns vary, the arithmetic mean is higher than the geometric mean (for returns greater than −100%). Using the arithmetic mean as if it were a compound growth rate overstates what a buy-and-hold path delivered.

Worked two-year path. Year 1: +50%. Year 2: −50%. Start at $100.

• End of year 1: $150. End of year 2: $75.
• Arithmetic mean = (0.50 + (−0.50)) / 2 = 0%. A 0% compound rate would have left you at $100. You do not have $100.
• Geometric mean = √(1.50 × 0.50) − 1 = √0.75 − 1 ≈ −13.4%. A constant −13.4% per year for two years is what actually happened to wealth: 100 × (0.866)² ≈ 75.

That is why geometric means matter to investors and to technicians who quote average performance of a signal, a market, or a multi-year trend. If the question is ending wealth or compound growth, geometric is the mean that matches the equity curve. If the question is the expected value of one random period drawn from the same distribution, arithmetic is the mean that matches that bet. Mixing the two is a standard Level I trap.

Equal returns have equal geometric and arithmetic means: three years of +10% compound at 10%. Variability is what drives the wedge. Higher volatility, wider wedge—another reason dispersion is not optional decoration.

Measures of dispersion

Dispersion (spread, variation) describes how far observations sit from the center. Two samples can share a mean of 0% daily and still be different markets: one with daily moves of 0.2%, one with daily moves of 3%.

Common measures:

• Range = maximum − minimum. Easy, fragile: one outlier is the range.
• Interquartile range (IQR) = Q3 − Q1, the width of the middle 50%. Used in the next section to fence outliers.
• Mean absolute deviation = average of |xᵢ − x̄|. Intuitive, less used in exam formulas than variance.
• Variance and standard deviation, the workhorses.

Variance and standard deviation

Variance is the average of squared deviations from the mean. Squaring does two jobs: it makes every deviation positive, and it penalizes large misses more than small ones. Population variance σ² divides by N. Sample variance s² divides by n − 1 (Bessel's correction) so that s² is an unbiased estimator of σ² when the draws are well behaved. CMT stems that say "sample" want n − 1 unless they give you the population formula.

Standard deviation is the square root of variance. It returns to the same units as the data (percent, dollars, points). That is why technicians quote SD, not variance, as a volatility number. Variance is the algebra underneath; SD is the ruler.

Tiny sample. Three period returns: 1%, 3%, 5%. Sample mean = 3%. Deviations: −2, 0, +2 percentage points. Squared: 4, 0, 4. Sample variance s² = (4 + 0 + 4) / (3 − 1) = 4. Sample SD s = 2 percentage points. Population variance using N = 3 would be 8/3 ≈ 2.67, SD ≈ 1.63. The stem must say sample or population.

A larger SD means a wider typical scatter around the mean. It does not tell you direction. A market can have a high SD while the mean return is still near zero: violent two-sided noise. Another market can trend with a modest SD if most deviations are small and one-sided. Read SD with the mean, not instead of it.

Z-scores and what they are for

A z-score (standard score) locates an observation in SD units from the mean:

z = (x − mean) / SD.

Using the tiny sample, a new return of 7% against mean 3% and s = 2% has z = (7 − 3) / 2 = +2. The print sits two sample standard deviations above the mean of that window.

Uses on Level I:

1. Standardize. A +2% day is huge for a 0.4% SD name and routine for a 2% SD name. Z-scores make those days comparable.
2. Flag unusual observations. Many rules of thumb treat |z| > 2 or |z| > 3 as unusual. The next section's outlier fences are a different rule; z is the standardized version.
3. Bridge to probability. If returns were roughly normal, a z of 2 is a 2-sigma event (Empirical Rule: about 95% of observations inside ±2 SD). Section 9.3 will warn you that real returns are not reliably Gaussian, so the probability translation is approximate.

Z-scores inherit every flaw of the mean and SD you plugged in. A window that includes a crash has a swollen SD; later ordinary days look small on z. A window that is too quiet makes a normal news day look like a +3 event. Name the sample.

Putting the toolkit on a Level I stem

When a question gives a short list of prices or returns:

1. Say whether the data are categorical or numerical.
2. Compute or identify mean, median, mode if the center is at issue; watch the outlier.
3. If the story is compound wealth, reach for the geometric mean, not the arithmetic mean.
4. If the story is typical scatter, quote SD (and know that variance is SD squared).
5. If the story is how unusual is this print, compute a z-score.

That sequence is the whole of Part 1. Part 2 adds pictures, outliers, scatterplots, and the line that tries to predict y from x.

Key Takeaways

• Statistics: describe a sample, or infer from it toward a population, with uncertainty
• Two data types: qualitative/categorical versus quantitative/numerical
• Mean, median, mode locate center; mean follows outliers
• Geometric mean matches compound wealth; arithmetic mean does not when returns vary
• Variance is mean squared deviation; SD is its square root; z = (x − mean) / SD