Z-Scores and Standardization

A z-score tells you how many standard deviations a value is from the mean. It's a powerful tool that lets you compare props across completely different stats and players.

The Z-Score Formula

Where:

• x = the line you're evaluating
• μ = mean (average)
• σ = standard deviation

Example: Converting to Z-Score

Player averages: μ = 25 points, σ = 8 points
Line: 33 points

z = (33 - 25) / 8 = 1.0

A z-score of 1.0 means the line is exactly 1 standard deviation above the mean.

Interpreting Z-Scores

Quick Assessment Rule

• z < 0: Line is below average → Over is more likely ("easy" line)
• z = 0: Line equals average → 50/50 proposition
• z > 0: Line is above average → Under is more likely ("hard" line)

Why Z-Scores Are Useful for Betting

1. Compare Across Different Props

A line 1σ above the mean is equally difficult whether it's points, yards, or rebounds.

All three props have the same z-score, so all three have roughly the same probability of hitting the over—despite being completely different stats.

2. Quick Line Assessment

You can instantly gauge if a line is "easy" or "hard":

z = (line - your_mean) / your_stdev

• If z < 0: The line is set below your projection. Over has value.
• If z > 0: The line is set above your projection. Under has value.
• The further from 0, the bigger the potential edge.

3. Line Shopping Comparison

Z-scores help compare the same prop across different sportsbooks:

The lower the z-score, the better for over bets. The higher the z-score, the better for under bets.

Using Z-Scores with Excel

Method 1: Calculate Z-Score, Then Use Standard Normal

Step 1: =(@line - mean) / stdev         → gives z-score
Step 2: =1-NORM.S.DIST(z, TRUE)         → gives P(Over)

Method 2: Direct Calculation (Equivalent)

=1-NORM.DIST(line, mean, stdev, TRUE)

Both methods give identical results. Use whichever feels more intuitive.

Confidence Intervals

Z-scores also help you construct confidence intervals—ranges where you expect the outcome to fall.

Example: LeBron's 95% Confidence Interval

μ = 25.7, σ = 6.7

95% range = 25.7 ± (1.96 × 6.7)
          = 25.7 ± 13.1
          = 12.6 to 38.8 points

You'd expect LeBron to score between 12.6 and 38.8 points in 95% of games.

Finding a Specific Percentile with NORM.INV

Sometimes you want to work backwards: "What point total does LeBron exceed 75% of the time?"

Example: What point total did LeBron exceed 75% of the time in 2023-24?

=NORM.INV(0.25, 25.7, 6.7)

Result: 21.2 points

LeBron scored more than 21.2 points in 75% of his games.

Quick Reference: Z-Score to Probability