Normal Probability Calculator for Sampling Distributions

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The true average of the entire population. For example, if the average weight of all adults in a country is 70 kg, enter 70.
How spread out individual values are in the population. Must be greater than zero.
The number of observations in your random sample. Larger samples produce a tighter sampling distribution.
Choose whether you want the probability that the sample mean is between two bounds, below a bound, or above a bound.
The lower bound for a between calculation, or the single cutoff for a left-tailed or right-tailed calculation.
The upper bound for a between calculation. Must be greater than X1.
ProbabilityVery likely (> 95%)
0.99383

The probability that the sample mean satisfies your chosen condition.

Standard error (σ/√n)1.8257
Z-score of X1-2.7386
Z-score of X22.7386
Probability (%)0.9938%
-2.73860.3% below · Z-score

There is a 99.38% chance the sample mean falls between 45 and 55.

  • Standard error = 1.8257: each sample of 30 observations has a mean that varies by about this much from μ = 50.
  • X1 = 45 is 2.74 standard errors below the mean (z = -2.7386).
  • X2 = 55 is 2.74 standard errors above the mean (z = 2.7386).

Next stepWith n >= 30 the sampling distribution of the mean is approximately normal regardless of population shape, so this result is reliable.

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