The Truth:

This is chess data. A win = 1 point, a draw = 0.5 points.

So 2.5 means: 2 wins + 1 draw. You're counting games, not measuring height or temperature.

Counting = Discrete. Even with decimal points, you're still counting outcomes, not measuring a continuum.

Remember: if you're not measuring (heights, temps, weights), you're counting. Discrete ≠ only integers. Discrete = distinct, separate outcomes.

What this means:

• X_i = The streak value (e.g., 2 = two consecutive wins)
• F_i = Frequency (e.g., 72 players had a streak of 2 wins)
• So the dataset is: [2, 2, 2, ... (72 times), 2.5, 2.5, ... (9 times), 3, 3, ... (41 times), ...]

Note: X (without subscript) = entire dataset. X_i (with subscript) = individual value.

Table Headers & Meanings:

• X_i: Datapoint
• F_i: Absolute Frequency
• CumF_i: Cumulative Absolute Frequency
• f_i: Relative Frequency (F_i ÷ total)
• cumf_i: Cumulative Relative Frequency
• F_iX_i: For calculating mean
• |X_i - μ|: Absolute deviation from mean
• F_i|X_i - μ|: For Mean Absolute Deviation
• (X_i - μ)²: Squared deviation
• F_i(X_i - μ)²: For variance
• X_i²: Squared value
• F_iX_i²: Alternative variance calculation

Most of these are intermediate calculations you'll never look at again, but you need 'em to get the real stats.

Key Observations:

• Relative frequency (f_i) tells you what percentage of the total each value represents. 2 appears 16.86% of the time.
• Cumulative frequency shows running totals. By streak 5.5, you've seen 52.69% of all data.
• This table lets you calculate everything: mean, variance, skewness, kurtosis, percentiles...

The Problem:

You're given this table already aggregated. You don't know the individual values anymore.

Example: Interval [2-10] has 35 data points. Which 35 values? Fuck if I know. Could be 2, 3, 5.5, 9... anything in that range.

This is common in published data, surveys, or when someone already processed the data for you (probably poorly).

Column Definitions:

• L_inf, L_sup: Lower/upper class limits
• a_i: Class width = L_sup - L_inf
• C_i: Class midpoint = (L_inf + L_sup) ÷ 2
• h_i: Frequency density = f_i ÷ a_i (for histograms)
• H_i: Cumulative frequency density

Note: This is purely an academic exercise. In real life, why would you start with data already in a table? Because your professor is a sadist, that's why.

Example Data:

• N = 100 (sample size)
• Range = 8
• Q1 = 1, Q3 = 4, IQR = 3
• Standard deviation = 3

Which One to Use?

• Square Root: Simple, okay for small datasets
• Sturges: Classic, biased toward normal distributions
• Scott: Good for normal data, uses standard deviation
• Freedman-Diaconis: Robust, uses IQR (good for skewed data)

My advice: Try a few, see which gives meaningful intervals. Don't blindly follow formulas.

Key Differences:

• Mean: μ = Σ(f_i·C_i) where C_i = class midpoint
• Variance: σ² = Σ[f_i·(C_i - μ)²] (using midpoints)
• Percentiles/Median: Use linear interpolation within classes
• Mode: Use formula based on adjacent class frequencies

Percentile Calculation Example (Median):

P₅₀ = 0.5 (50th percentile)

cum f(Me-1) = 0.3500 (cumulative frequency before median class)

f(Me) = 0.2500 (frequency of median class)

li(Me) = 10.0000 (lower limit of median class)

a(Me) = 5.0000 (width of median class)

Formula: Me = li(Me) + [(0.5 - cum f(Me-1)) ÷ f(Me)] × a(Me)

Me = 10 + [(0.5 - 0.35) ÷ 0.25] × 5 = 13.0000

Graph Choice Depends on Table Type:

• Disaggregated discrete data: Bar charts, dot plots, stem-and-leaf
• Aggregated continuous data: Histograms, frequency polygons, ogives
• Mixed situations: Box plots work for both (using raw or binned data)

For our chess example (discrete): bar chart or stem-and-leaf. For the aggregated example: histogram or cumulative frequency graph.