Standard Deviation and Probability Basics

Standard Deviation

Two stores can have the same average delivery time but very different reliability. Store A: 4, 4, 4, 4, 4. Store B: 1, 2, 4, 6, 7. Both have a mean of 4, but Store B is unpredictable. Spread measures this.

Range = max − min. Store A: 0. Store B: 6. Simple, but it only uses two values.

Standard deviation is roughly the typical distance of each value from the mean. Step by step for Store B (mean 4):

1. Distance from mean: −3, −2, 0, 2, 3
2. Square each (so negatives do not cancel): 9, 4, 0, 4, 9
3. Average the squares (this is the variance). For a sample, divide by n − 1: 26 / 4 = 6.5
4. Take the square root: √6.5 ≈ 2.55 days

So deliveries at Store B typically land about 2.5 days away from the 4-day average. Store A's standard deviation is 0.

Excel: =STDEV.S(range) for a sample (the usual choice) and =STDEV.P(range) when you have the whole population. pandas: df["days"].std().

Interpreting it: a small standard deviation means values cluster tightly around the mean; a large one means they are scattered. Always report it next to the mean, in the same units.

Probability Basics

Probability is a number from 0 (impossible) to 1 (certain) describing how likely something is. It is often written as a percentage.

P(event) = number of ways it can happen / total number of equally likely outcomes

Rolling a 6 on a die: 1 / 6 ≈ 0.167. In business you usually estimate probability from data: if 45 of 900 website visitors bought something, P(purchase) = 45 / 900 = 0.05, a 5% conversion rate.

Key rules:

• Complement: P(not A) = 1 − P(A). If 5% buy, 95% do not.
• Or (events that cannot both happen): P(A or B) = P(A) + P(B). Rolling a 1 or a 2 = 1/6 + 1/6 = 1/3.
• And (independent events): P(A and B) = P(A) × P(B). Two coin flips both heads = 0.5 × 0.5 = 0.25. Events are independent when one does not change the chance of the other.

Conditional probability is the chance of A given that B happened, written P(A | B). Example: of 200 mobile visitors, 6 bought; of 100 desktop visitors, 8 bought. P(buy | mobile) = 3%, P(buy | desktop) = 8%. Comparing conditional probabilities like this is everyday analyst work.