std & var: Measuring Consistency

Two batters can have exactly the same average and still be completely different players. One scores 48–52 every match; the other scores 0, 10, 100, 5, 135. The average hides this. To measure consistency you need a measure of spread — and the two standard ones are variance and standard deviation.

Definition

Variance (var) is the average of the squared distances of each value from the mean. Standard deviation (std) is the square root of the variance, which brings the spread back into the original units (runs). A low std means values sit close to the mean (consistent); a high std means they are scattered (inconsistent).

• Distance from the mean — for each score, score - mean. Some are positive, some negative.
• Squared — squaring makes every distance positive and punishes big misses more.
• Square root — variance is in "runs²", which is hard to picture; std = sqrt(var) is back in runs.

The formulas (for n values with mean μ):

variance  = sum( (x - μ)² ) / (n - ddof)
std       = sqrt(variance)

Syntax

np.std(a, axis=None, ddof=0, keepdims=False)     a.std(axis=None, ddof=0)
np.var(a, axis=None, ddof=0, keepdims=False)     a.var(axis=None, ddof=0)

Population vs sample — what is ddof?

Dividing by n - 1 makes the sample result slightly bigger, which corrects for the fact that a small sample usually underestimates the true spread.

Note: this is a classic surprise: NumPy's std() and Pandas' .std() give different answers on the same numbers, because NumPy defaults to ddof=0 and Pandas to ddof=1. Pass ddof explicitly when the difference matters.

Example — same average, different consistency

This is the "measuring consistency" example from the course notebook:

import numpy as np

player_a = np.array([50, 48, 52, 49, 51])
player_b = np.array([0, 10, 100, 5, 135])

print("Player A:", player_a)
print("  mean =", player_a.mean(), " std =", round(float(player_a.std()), 2))
print("Player B:", player_b)
print("  mean =", player_b.mean(), " std =", round(float(player_b.std()), 2))

Player A: [50 48 52 49 51]
  mean = 50.0  std = 1.41
Player B: [  0  10 100   5 135]
  mean = 50.0  std = 56.3

How it works: both means are 50.0. Player A's scores are within about 1–2 runs of 50, so std ≈ 1.41. Player B's scores are 50 runs away on average, so std ≈ 56.3. A captain picks A for a tight chase and B as a high-risk, high-reward gamble.

Example — computing it by hand (to see what std really is)

import numpy as np

x = np.array([50, 48, 52, 49, 51])
mean = x.mean()
deviations = x - mean
squared = deviations ** 2

print("deviations:", deviations)
print("squared   :", squared)
print("variance  :", squared.mean(), " np.var:", x.var())
print("std       :", np.sqrt(squared.mean()), " np.std:", x.std())
print("sample var (ddof=1):", x.var(ddof=1))
print("sample std (ddof=1):", x.std(ddof=1))

deviations: [ 0. -2.  2. -1.  1.]
squared   : [0. 4. 4. 1. 1.]
variance  : 2.0  np.var: 2.0
std       : 1.4142135623730951  np.std: 1.4142135623730951
sample var (ddof=1): 2.5
sample std (ddof=1): 1.5811388300841898

How it works: the squared deviations add to 10. Population variance divides by 5 → 2.0; sample variance divides by 4 → 2.5. The std is just the square root of each.

Example — consistency report for the squad

import numpy as np

players = ["Virat", "Rohit", "Hardik", "Shubman"]
scores = np.array([
    [72, 45, 88, 31, 64],
    [56,  0, 74, 88, 32],
    [45, 32,  0, 67, 18],
    [33, 61, 45,  0, 77],
])

means = scores.mean(axis=1)
stds = scores.std(axis=1)
for name, m, s in zip(players, means, stds):
    print(f"{name:<8} avg {m:5.1f}   std {s:5.1f}")

print("Most consistent :", players[stds.argmin()])
print("Least consistent:", players[stds.argmax()])

Virat    avg  60.0   std  20.0
Rohit    avg  50.0   std  31.2
Hardik   avg  32.4   std  22.9
Shubman  avg  43.2   std  26.2
Most consistent : Virat
Least consistent: Rohit

How it works: axis=1 gives one std per player (row). argmin/argmax return the position of the smallest/largest std, which we use to look up the player's name.

Common mistakes

• ❌ Judging players only by average. ✅ Same mean does not mean same behaviour — always check std alongside the mean.
• ❌ Expecting NumPy and Pandas std to match. ✅ NumPy uses ddof=0, Pandas uses ddof=1; set it explicitly.
• ❌ Comparing a variance with a mean ("var is 800, mean is 50"). ✅ Variance is in squared units; compare the std with the mean.
• ❌ a.std(ddof=1) on a single value → nan plus a warning (divide by 0). ✅ A sample std needs at least 2 values.

Comparison — var vs std

Interview tip

Q: What is the difference between np.std(x) and np.std(x, ddof=1)? A: The first is the population standard deviation (divides by n); the second is the sample standard deviation (divides by n − 1, Bessel's correction). Use ddof=1 when your data is a sample from a larger population — it's also what Pandas and most statistics textbooks use by default.