Confidence Interval Calculator

Confidence Interval is evaluated from Sample Mean or Proportion, Sample Standard Deviation and Sample Size. The calculation reports Margin of Error, Lower Bound and Upper Bound.

Results

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About the Confidence Interval Calculator

Confidence Interval is treated here as a quantitative relation between Sample Mean or Proportion, Sample Standard Deviation, Sample Size and Confidence Level and Margin of Error, Lower Bound, Upper Bound and Standard Error.

The calculator uses a multi formula configuration. Each reported value is read as a direct evaluation of the stored rules with the declared field formats and units.

Formula basis:
SE = s / sqrtn
Margin of Error E = z* x SE
95% CI: (x̄ - E, x̄ + E)

Interpret the outputs in the order shown by the result fields. Optional inputs affect only the outputs that depend on those variables.

Formula & How It Works

The calculation applies the following relations exactly as recorded in the metadata:

SE = s / sqrtn
Margin of Error E = z* x SE
95% CI: (x̄ - E, x̄ + E)

Each output field is produced by substituting the supplied inputs into the relevant relation and then applying the declared rounding or text format.

Worked Examples

Example 1: Customer Satisfaction Survey

Inputs

x_bar: 7.8 sd_val: 1.5 n_val: 100 conf: 0.95
Margin of Error: 0.294. Lower Bound: 7.506. Upper Bound: 8.094. Standard Error: 0.15

With Sample Mean or Proportion = 7.8, Sample Standard Deviation = 1.5, Sample Size = 100 and Confidence Level = 0.95 as the stated inputs, the result is Margin of Error = 0.294, Lower Bound = 7.506 and Upper Bound = 8.094. Each value corresponds to the declared output fields.

Example 2: Election Poll Margin of Error

Inputs

x_bar: 0.53 sd_val: 0.499 n_val: 1000 conf: 0.95
Margin of Error: 0.0309. Lower Bound: 0.4991. Upper Bound: 0.5609. Standard Error: 0.01578

With Sample Mean or Proportion = 0.53, Sample Standard Deviation = 0.499, Sample Size = 1,000 and Confidence Level = 0.95 as the stated inputs, the result is Margin of Error = 0.0309, Lower Bound = 0.4991 and Upper Bound = 0.5609. Each value corresponds to the declared output fields.

Example 3: Mean Commute Time — Wider CI

Inputs

x_bar: 32.5 sd_val: 8.2 n_val: 35 conf: 0.99
Margin of Error: 3.5705. Lower Bound: 28.9295. Upper Bound: 36.0705. Standard Error: 1.386053

With Sample Mean or Proportion = 32.5, Sample Standard Deviation = 8.2, Sample Size = 35 and Confidence Level = 0.99 as the stated inputs, the result is Margin of Error = 3.5705, Lower Bound = 28.9295 and Upper Bound = 36.0705. Each value corresponds to the declared output fields.

Example 4: Drug Efficacy — Blood Pressure Reduction

Inputs

x_bar: 12.4 sd_val: 4.1 n_val: 64 conf: 0.95
Margin of Error: 1.0045. Lower Bound: 11.3955. Upper Bound: 13.4045. Standard Error: 0.5125

With Sample Mean or Proportion = 12.4, Sample Standard Deviation = 4.1, Sample Size = 64 and Confidence Level = 0.95 as the stated inputs, the result is Margin of Error = 1.0045, Lower Bound = 11.3955 and Upper Bound = 13.4045. Each value corresponds to the declared output fields.

Common Use Cases

  • Find 95% CI for a sample mean
  • Calculate margin of error for a survey
  • Estimate population proportion with confidence bounds
  • Determine required sample size for given margin