Normal Distribution Calculator

Three boxes, pre-filled with x = 168, mean = 170, SD = 6.28. Click Calculate. The normal distribution calculator form asks for one value x plus the two numbers that define the bell: mean μ and standard deviation σ.

• x is the value you care about — a height, a score, a measurement.
• μ centers the bell; σ sets its width. Together they say X ~ N(μ, σ).
• Three answers come back: the z-score of x, the left-tail probability P(X < x), and the right-tail probability P(X > x). Fill both optional interval boxes a and b and the normal distribution calculator adds the between-values row P(a < X < b).

The normal distribution calculator keeps full precision, so 168 against 170 and 6.28 returns z ≈ −0.32 and a percentile near 37.5%.

This page is more than a static normal distribution calculator — it is also an AI tutor you can ask questions:

• Ask in plain words. "Heights average 170 with SD 6.28 — what fraction is under 180?" The AI fills the form and runs this normal distribution calculator for you.
• Take a photo of your homework. The AI reads the distribution and the question, then fills the normal distribution calculator boxes for you.
• Ask about the ideas, not just the answer. "What does the z-score mean?" The AI explains step by step — every number still comes from the normal distribution calculator, so nothing is guessed.
• Keep asking follow-ups. Ask "why?" as often as you need — follow-up questions are welcome.
• Between-values and inverse questions. P(a < X < b) and "find the 95th percentile" start here too; the AI walks the same bell.

Heights pile up. Most men stand near the average; very tall and very short are rare. Plot a huge sample of heights and the histogram traces a bell the normal distribution calculator can weigh. That shape is the normal distribution, and this normal distribution calculator works on it. The example data is male heights averaging μ = 170 cm with σ = 6.28 cm, so X ~ N(170, 6.28).

The picture carries the two facts that run every normal distribution calculator question. First, the curve is symmetric around μ. Second, spread is measured in σ steps: the shaded band from one σ below to one σ above the mean holds about 68% of all values, two σ holds 95%, three holds 99.7%. That is the 68-95-99.7 rule, and it is why one number z — how many standard deviations x sits from μ — answers every question about any bell.

Read the picture first: bell centered at μ, width in σ steps. Now the symbols the normal distribution calculator uses:

$z = \frac{x - \mu}{\sigma}$

Every number the normal distribution calculator returns is one of these three lines, applied to your x, μ, and σ. That is the whole normal distribution calculator.

Heights follow X ~ N(170, 6.28). Standardize x = 168 — the normal distribution calculator's bread and butter.

1. $z = (168 - 170)/6.28 \approx -0.32$ — the value sits 0.32 SDs below the mean.
2. Left-tail: $P(X < 168) = \Phi(-0.32) \approx 37.5\%$ — about 37.5% of men are shorter.
3. Right-tail: $P(X > 168) = 1 - 0.375 = 62.5\%$.

Between 165 and 175? That is z = ±0.80, so P(165 < X < 175) = 2(0.787) − 1 ≈ 57.4%. Now run the normal distribution calculator backwards: a z of 1.27 corresponds to $x = 170 + 1.27(6.28) \approx 177.98$ — the reverse trip the normal distribution calculator's z row starts.

A weight-loss program gives the normal distribution calculator a second bell: results X ~ N(5, 2), in pounds. Standardize x = 10 and x = −3.

1. Ten pounds lost: $z = (10 - 5)/2 = 2.5$ — "2.5 standard deviations to the right of the mean."
2. Three pounds gained (x = −3): $z = (-3 - 5)/2 = -4$ — four SDs the other way.
3. The sign carries the direction; the normal distribution calculator keeps it automatically.

Different units, different bells — same z scale. That is the point of standardizing: the normal distribution calculator compares heights with weight loss on one scale, with zero unit conversion.

Test scores hand the normal distribution calculator a third bell: μ = 50, σ = 6. Use the 68-95-99.7 rule for a change — no normal distribution calculator needed at this step.

1. 68% of scores fall between $50 - 6 = 44$ and $50 + 6 = 56$.
2. 95% fall between $50 - 12 = 38$ and $50 + 12 = 62$.
3. 99.7% fall between 32 and 68 — almost nobody outside.

For anything finer than the rule of thumb, type 56 into the x box of the normal distribution calculator above; the left-tail row hands you the exact answer.

Four slips the normal distribution calculator catches every day:

1. Looking up a raw x on the standard normal table. The standard curve has μ = 0 and σ = 1; your x must become z first. The normal distribution calculator does that division before anything else.
2. Mixing up the two tails. Tables and the P(X < x) row give the left-tail area; "more than x" is the complement, 1 − left. The normal distribution calculator prints both so you never subtract in your head.
3. Dropping the minus sign. When x is below μ, z is negative. Forgetting the sign flips a 37.5% answer into 62.5% — the exact mistake the normal distribution calculator's z row exposes.
4. Reading percentile backwards. The percentile of x is the area to its LEFT, the fraction of values below it. Not the z-score itself, and not the right tail.