Source: Mid-Term Formula Sheet, STAT 601 (UCF)
Tags: descriptive statistics, mean, variance, standard deviation, covariance, correlation, skewness, box plot, IQR, percentile, z-score, Chebyshev, empirical rule, STAT 601
Difficulty: Foundational Prerequisites: Basic algebra, summation notation.
This is where every statistics course begins: how do you take a pile of numbers and say something useful about them? These chapters give you the vocabulary (what "statistics" even means, what DCOVA stands for) and the core toolkit for summarising a dataset with a single number (mean, median) or describing its spread (variance, standard deviation). You also learn to spot shape (skewness), identify outliers (box plots, z-scores), and measure how two variables move together (covariance, correlation). Everything in later chapters, from probability distributions to hypothesis testing, builds directly on these measures. If you are shaky here, the rest of the course will feel harder than it needs to.
Statistics turns raw data into decisions. Chapters 1–3 cover the measures you will use constantly: means and medians for centre, variance and standard deviation for spread, covariance and correlation for relationships between variables, and z-scores for standardising values so you can compare across different scales.
Statistics
The methods that help transform data into useful information for decision makers.
DCOVA
Define, Collect, Organise, Visualise, Analyse. The standard process for working with data. The goals are to summarise, conclude, forecast or predict, and improve.
Business analytics
A combination of statistics, management science/operations research, and information systems, used to reach business decisions. Sometimes involves "Big Data."
Big Data
Data characterised by high volume, velocity, and/or variety. In simple terms, datasets so large or fast-moving that traditional tools struggle with them.
Population
The complete set of all items of interest. Parameters describe populations (Greek letters: μ, σ).
Sample
A subset drawn from the population. Statistics describe samples (Roman letters: x̄, s).
Mean (average)
The sum of all values divided by how many there are. Sample mean: x̄ = Σxᵢ / n. Population mean: μ = Σxᵢ / N. Think of it as the balance point of the data.
Variance
The average of the squared deviations from the mean. Sample: s² = Σ(xᵢ − x̄)² / (n − 1). Population: σ² = Σ(xᵢ − μ)² / N. It measures how spread out values are, but in squared units.
Standard deviation
The square root of variance. Sample: s = √[Σ(xᵢ − x̄)² / (n − 1)]. Population: σ = √[Σ(xᵢ − μ)² / N]. In simple terms, this puts the spread back into the original units so it is easier to interpret.
Coefficient of variation (CV)
CV = |s / x̄| × 100% (sample) or |σ / μ| × 100% (population). Expresses spread as a percentage of the mean, which lets you compare variability across datasets with different scales or units. Only meaningful when the mean is not near zero.
Covariance
A measure of how two variables move together. Sample: cov(X, Y) = Σ(xᵢ − x̄)(yᵢ − ȳ) / (n − 1). Positive means they tend to increase together; negative means one tends to decrease as the other increases. The raw number is hard to interpret on its own because it depends on the units.
Correlation coefficient (r)
r = cov(X, Y) / (Sₓ · Sᵧ). Ranges from −1 to +1. Strips out the units problem from covariance, giving a standardised measure of linear association. +1 is a perfect positive line, 0 is no linear relationship, −1 is a perfect negative line. The population version uses ρ (rho) and follows the same interpretation.
Percentile
The value below which a given percentage of the data falls. Position: i = percentile × n. If i is not an integer, round up. If i is an integer, average the values at positions i and i + 1.
Median
The 50th percentile. The middle value when data is sorted. Sometimes written x̃.
Interquartile range (IQR)
IQR = Q₃ − Q₁. The range of the middle 50% of the data. Resistant to outliers, unlike the full range.
Z-score (z-value)
z = (xᵢ − x̄) / σ (population σ known) or t = (xᵢ − x̄) / s (σ unknown). Tells you how many standard deviations a value sits from the mean. A z of −3 means three standard deviations to the left.
Sample mean: x̄ = Σxᵢ / n
Population mean: μ = Σxᵢ / N
Median: the 50th percentile, robust to extreme values
Midrange: (x_min + x_max) / 2 (rarely used, highly sensitive to outliers)
Midhinge: (Q₁ + Q₃) / 2
Variance and standard deviation (see Key Terms above)
Coefficient of variation for comparing spread across different scales
IQR = Q₃ − Q₁
MAD (Mean Absolute Deviation): Σ|xᵢ − x̄| / n
Covariance captures direction of the relationship but is scale-dependent
Correlation (r or ρ) standardises covariance to [−1, +1]
Both measure linear association only; a correlation of 0 does not mean "no relationship," just no linear one
Upper Inner fence: Q₃ + 1.5(IQR)
Lower Inner fence: Q₁ − 1.5(IQR)
Upper Outer fence: Q₃ + 3(IQR)
Lower Outer fence: Q₁ − 3(IQR)
Points beyond inner fences are mild outliers; beyond outer fences, extreme outliers
Outer fences are not always shown in practice
Left skew: mean < median (tail stretches left)
No skew: mean ≈ median
Right skew: mean > median (tail stretches right)
This rule works best for approximately unimodal continuous distributions. It often fails for discrete data, so use software for an exact skew measure when in doubt.
Empirical Rule (requires a normal distribution):
μ ± 1σ contains about 68.26% of values
μ ± 2σ contains about 95.44%
μ ± 3σ contains about 99.73%
Chebyshev's Theorem (works for any distribution):
μ ± 2σ contains at least 75%
μ ± 3σ contains at least 88.9%
μ ± kσ contains at least (1 − 1/k²) × 100%
Scenario | σ known | σ not known |
|---|---|---|
Sample | zᵢ = (xᵢ − x̄) / σ | tᵢ = (xᵢ − x̄) / s |
Population | zᵢ = (xᵢ − μ) / σ | N/A |
Geometric mean: (x₁ · x₂ · … · xₙ)^(1/n). Used with products and rates of change.
Geometric mean rate of return: [(1 + R₁)(1 + R₂)…(1 + Rₙ)]^(1/n) − 1
Average growth for a time series: (xₙ / x₁)^(1/(n−1)) − 1
All summations run over all i unless noted.
Measure | Sample | Population |
|---|---|---|
Mean | x̄ = Σxᵢ / n | μ = Σxᵢ / N |
Variance | s² = Σ(xᵢ − x̄)² / (n − 1) | σ² = Σ(xᵢ − μ)² / N |
Std Dev | s = √(s²) | σ = √(σ²) |
CV | |s / x̄| × 100% | |σ / μ| × 100% |
Covariance | Σ(xᵢ − x̄)(yᵢ − ȳ) / (n − 1) | Σ(xᵢ − μₓ)(yᵢ − μᵧ) / N |
Correlation | r = cov(X,Y) / (Sₓ · Sᵧ) | ρ = cov(X,Y) / (σₓ · σᵧ) |
Histogram class width: w = Range (rounded up) / number of classes.
The coefficient of variation shows up constantly in finance: comparing the risk (volatility) of two investments that have very different average returns. A CV of 20% on one fund versus 50% on another tells you far more than comparing raw standard deviations in different currencies or scales.
Z-scores are the foundation of quality control. When a manufacturing process produces a part whose z-score exceeds ±3, the machine is likely drifting out of spec and needs recalibration.
Students often confuse n − 1 (sample variance denominator) with N (population variance denominator). The n − 1 corrects for bias when estimating population variance from a sample. If you are computing the variance of an entire population, use N.
A correlation near zero does not mean "no relationship." It means no linear relationship. Two variables can have a strong curved pattern and still show r ≈ 0.
Students sometimes apply the Empirical Rule to any dataset. It only applies to approximately normal distributions. For non-normal data, Chebyshev's Theorem is the safe fallback.
Skewness direction trips people up: in right-skewed data, the tail goes to the right and the mean is pulled to the right of the median, not the other way around.
⚠️ Know when to use n − 1 versus N in the denominator. This comes up in nearly every calculation.
⚠️ Be able to interpret r: its sign, its magnitude, and what it cannot tell you (causation, non-linear relationships).
⚠️ Chebyshev's vs. Empirical Rule: know which requires normality and which works for any shape.
⚠️ Box plot fences: be able to compute inner and outer fences and identify outliers from them.
⚠️ Z-scores: comfortable converting between raw values and standard deviations, in both directions.
True or False: The sample variance divides by n, not n − 1.
Fill in the blank: A correlation of −1 indicates a ________ linear relationship.
True or False: Chebyshev's Theorem guarantees at least 68% of data falls within one standard deviation of the mean.
Fill in the blank: If the mean is greater than the median, the distribution is ________ skewed.
True or False: The coefficient of variation is useful for comparing spread across datasets with different units.
Answers: 1. False (sample variance uses n − 1). 2. Perfect negative. 3. False (Chebyshev gives ≥ 0% for k = 1; the 68% figure is the Empirical Rule for normal distributions). 4. Right. 5. True.
Q: A dataset has Q₁ = 20 and Q₃ = 40. What is the upper inner fence?
A: IQR = 40 − 20 = 20. Upper inner fence = Q₃ + 1.5 × IQR = 40 + 30 = 70.
Q: A sample of five values has a mean of 10. You discover one value was recorded as 15 instead of 5. What is the corrected mean?
A: Original sum = 10 × 5 = 50. Corrected sum = 50 − 15 + 5 = 40. Corrected mean = 40 / 5 = 8.
Q: Two variables have cov(X,Y) = 12, Sₓ = 3, and Sᵧ = 8. What is the correlation?
A: r = 12 / (3 × 8) = 12 / 24 = 0.5. A moderate positive linear relationship.
Q: Using Chebyshev's Theorem, at least what percentage of data falls within 4 standard deviations of the mean?
A: (1 − 1/4²) × 100% = (1 − 1/16) × 100% = 93.75%.
Q: When should you use a t-value instead of a z-value for standardisation?
A: When the population standard deviation (σ) is unknown and you must use the sample standard deviation (s) instead.
These summary measures feed directly into the sampling distributions of Chapters 7a and 7b: the sample mean x̄ and sample proportion p become random variables with their own means and standard deviations. Variance and covariance reappear in portfolio analysis (Chapter 5) and in the pooled-variance t-test (Chapter 10). Mastering the z-score here makes the transition to hypothesis testing (Chapter 9) feel like a natural extension rather than a new concept.
descriptive statistics, sample mean, population mean, variance, standard deviation, coefficient of variation, covariance, Pearson correlation, r value, rho, percentile, quartile, median, IQR, interquartile range, box plot, whisker plot, outlier, fence, skewness, left skew, right skew, Empirical Rule, 68-95-99.7 rule, Chebyshev's theorem, Tchebysheff, z-score, z-value, t-value, standardisation, geometric mean, MAD, STAT 601, UCF