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Standard Error of the Mean (SE = s / √n) Calculator

By the Central Limit Theorem, the Standard Error of the Mean (SE) measures the sampling variability of sample means x̄ across repeated samples of size n: SE = s / √n.

SE = s / √n
Pick which value you want to calculate (or type into any field):Solving for: SE — Standard Error of the Mean
Calculated

Standard deviation of the sampling distribution of x̄

0.0011,000.00
unit

Observed dispersion of individual observations

0.0110,000.00
unit

Number of independent observations in the sample

2100,000.00
unit
Calculated Result · Standard Error of the Mean (SE)
1.5

Step-by-Step Calculation

01Canonical Law: SE = s / √n
02Given Inputs (Base Units): s = 15, n = 100
03Solve for SE (Standard Error of the Mean): SE = 1.5

By the Central Limit Theorem, the Standard Error of the Mean (SE) measures the sampling variability of sample means x̄ across repeated samples of size n: SE = s / √n.

Real-World Reference Values

Square-Root Law of Precision4× Sample Size → ½ SE

Halving standard error requires quadrupling sample size n

10× Precision Target100× Sample Size Required

Diminishing returns at very large n

95% Confidence Interval Half-WidthMOE ≈ 1.96 × SE

Links standard error directly to margin of error

20-Row Comparison Table

Standard Error of the Mean (SE) · Sample Standard Deviation (s)

Standard Error of the Mean (SE = s / √n) Calculator — Sample Standard Deviation → Standard Error of the Mean
#Sample Standard Deviation (s)Standard Error of the Mean (SE)Δ %
0115 Current1.5 0.00%
0273.1673 7.3167 +387.78%
03146.3246 14.6325 +875.50%
04219.4819 21.9482 +1363.21%
05292.6392 29.2639 +1850.93%
06365.7965 36.5796 +2338.64%
07438.9538 43.8954 +2826.36%
08512.1111 51.2111 +3314.07%
09585.2684 58.5268 +3801.79%
10658.4257 65.8426 +4289.50%
11731.583 73.1583 +4777.22%
12804.7403 80.474 +5264.94%
13877.8976 87.7898 +5752.65%
14951.0548 95.1055 +6240.37%
151,024.21 102.4212 +6728.08%
161,097.37 109.7369 +7215.80%
171,170.53 117.0527 +7703.51%
181,243.68 124.3684 +8191.23%
191,316.84 131.6841 +8678.94%
201,390.00 138.9999 +9166.66%
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Frequently Asked Questions

What is the difference between Standard Deviation (s) and Standard Error (SE)?+

Standard deviation s measures the spread of individual data points around the mean; Standard Error SE measures the precision of the estimated sample mean itself.

How many observations n do I need to reduce my Standard Error to 0.5?+

Click SOLVE FOR on n (Sample Size) and set SE = 0.5. The engine evaluates n = (s / SE)².

Why does Standard Error shrink according to √n instead of n?+

Variances of independent random variables add linearly (Var(Σx_i) = nσ²), so dividing the sum by n divides variance by n and standard deviation by √n.