Margin of Error in Statistics

Study Notes

Margin of Error

The margin of error is a measure of the amount of random sampling error in a survey's results.

Confidence Interval

A range of values within which the true population parameter is likely to lie
Calculated using the sample mean, sample size, and standard error
Typically expressed as a percentage, e.g., +/- 3%
The level of confidence (e.g., 95%) indicates the probability that the true population parameter lies within the interval

Sample Size

The number of respondents or data points in a sample
Affects the margin of error: larger samples yield smaller margins of error
Formula to calculate the sample size required for a desired margin of error:

Sample Size = (Z^2 \* σ^2) / E^2

Where: + Z = Z-score corresponding to the desired confidence level + σ = population standard deviation (estimated or known) + E = desired margin of error

Standard Error

A measure of the variability of the sample mean
Calculated as the sample standard deviation divided by the square root of the sample size
Formula:

Standard Error = σ / √n

Where: + σ = sample standard deviation + n = sample size

The standard error is used to construct the confidence interval and calculate the margin of error

Margin of Error

Measure of random sampling error in a survey's results
Affects the precision of the survey's results

Confidence Interval

Range of values within which the true population parameter is likely to lie
Calculated using the sample mean, sample size, and standard error
Expressed as a percentage (e.g., +/- 3%)
Level of confidence (e.g., 95%) indicates the probability that the true population parameter lies within the interval

Sample Size

Number of respondents or data points in a sample
Affects the margin of error: larger samples yield smaller margins of error
Formula to calculate the sample size required for a desired margin of error:
- Sample Size = (Z^2 * σ^2) / E^2
- Where: Z = Z-score corresponding to the desired confidence level
- σ = population standard deviation (estimated or known)
- E = desired margin of error

Standard Error

Measure of the variability of the sample mean
Calculated as the sample standard deviation divided by the square root of the sample size
Formula:
- Standard Error = σ / √n
- Where: σ = sample standard deviation
- n = sample size
Used to construct the confidence interval and calculate the margin of error

Confidence Interval

Margin of Error

Margin of error is the maximum difference between the sample statistic and the true population parameter
It measures the random sampling error in survey results
Formula: Margin of Error = (Z-score × Standard Error)
Expressed as a percentage, with smaller margins indicating more accurate estimates

Confidence Level

The probability that the confidence interval contains the true population parameter
Common levels: 95% (most commonly used), 99%, 90%
Higher confidence levels require larger sample sizes for the same margin of error
Formula: Confidence Level = 1 - α (alpha level), where α is the probability of Type I error
A 95% confidence level means 95% of repeated studies' confidence intervals would contain the true population parameter
Confidence level and margin of error are inversely related: increasing the confidence level increases the margin of error, and vice versa