The Margin of Error Calculator helps you determine the precision of your statistical estimates from sample data. It quantifies the range of values within which the true population parameter is likely to fall. Whether you're analyzing survey results, conducting research, or interpreting polls, this tool helps you understand the reliability of your estimates and report results with appropriate confidence intervals.
Use 50% for most conservative estimate
Enter values and click calculate to see the margin of error.
For a sample proportion of 50% with sample size 400, the margin of error tells you how much the sample estimate might differ from the true population proportion.
Margin of error (MOE) represents the maximum expected difference between your sample estimate and the true population parameter. It quantifies the uncertainty associated with sampling and provides a range within which the true population value is likely to fall.
For example, if a political poll shows that 52% of voters support a particular candidate with a margin of error of ±3%, this means the actual level of support in the entire population likely falls between 49% and 55%.
Given:
Result: The margin of error is ±4.49%
When sampling from a known, finite population, you can apply the finite population correction:
Given:
Result: The required sample size is 323 (rounded up)
When sampling from a known, finite population, you can apply the finite population correction:
When the expected proportion is unknown, use p = 0.5 (50%) to calculate the most conservative (largest) sample size:
Example for 95% confidence level and 5% margin of error:
Result: The maximum required sample size is 385 (rounded up)
If a survey finds that 60% of respondents favor a policy with a margin of error of ±3% at the 95% confidence level:
"We are 95% confident that between 57% and 63% of the population favors the policy."
import numpy as np
from scipy import stats
def margin_of_error_proportion(p, n, confidence_level=0.95, population_size=None):
z = stats.norm.ppf((1 + confidence_level) / 2)
se = np.sqrt(p * (1 - p) / n)
moe = z * se
if population_size and population_size > n:
fpc = np.sqrt((population_size - n) / (population_size - 1))
moe *= fpc
return moe * 100
def margin_of_error_mean(std_dev, n, confidence_level=0.95, population_size=None):
z = stats.norm.ppf((1 + confidence_level) / 2)
se = std_dev / np.sqrt(n)
moe = z * se
if population_size and population_size > n:
fpc = np.sqrt((population_size - n) / (population_size - 1))
moe *= fpc
return moe
p = 0.7
n = 400
moe_prop = margin_of_error_proportion(p, n, 0.95)
print(f"Margin of Error for proportion: ±{moe_prop:.2f}%")
# Margin of Error for proportion: ±4.49%
std_dev = 15
n = 100
moe_mean = margin_of_error_mean(std_dev, n, 0.95)
print(f"Margin of Error for mean: ±{moe_mean:.2f}")
# Margin of Error for mean: ±2.94
#---------------------------------------------------#
def sample_size_for_proportion(p, desired_moe, confidence_level=0.95, population_size=None):
desired_moe_decimal = desired_moe / 100
z = stats.norm.ppf((1 + confidence_level) / 2)
n = (z**2 * p * (1-p)) / (desired_moe_decimal**2)
if population_size and population_size > 0:
n = (n * population_size) / (n + population_size - 1)
return int(np.ceil(n))
def sample_size_for_mean(std_dev, desired_moe, confidence_level=0.95, population_size=None):
z = stats.norm.ppf((1 + confidence_level) / 2)
n = (z**2 * std_dev**2) / (desired_moe**2)
if population_size is not None and population_size > 0:
n = (n * population_size) / (n + population_size - 1)
return int(np.ceil(n))
moe = 5
p = 0.5
n = sample_size_for_proportion(p, moe)
print(f"Sample size for proportion with {moe}% margin of error: {n}")
# Sample size for proportion with 5% margin of error: 385
std_dev = 15
moe = 3
n = sample_size_for_mean(std_dev, moe)
print(f"Sample size for mean with {moe} margin of error: {n}")
# Sample size for mean with 3 margin of error: 97The Margin of Error Calculator helps you determine the precision of your statistical estimates from sample data. It quantifies the range of values within which the true population parameter is likely to fall. Whether you're analyzing survey results, conducting research, or interpreting polls, this tool helps you understand the reliability of your estimates and report results with appropriate confidence intervals.
Use 50% for most conservative estimate
Enter values and click calculate to see the margin of error.
For a sample proportion of 50% with sample size 400, the margin of error tells you how much the sample estimate might differ from the true population proportion.
Margin of error (MOE) represents the maximum expected difference between your sample estimate and the true population parameter. It quantifies the uncertainty associated with sampling and provides a range within which the true population value is likely to fall.
For example, if a political poll shows that 52% of voters support a particular candidate with a margin of error of ±3%, this means the actual level of support in the entire population likely falls between 49% and 55%.
Given:
Result: The margin of error is ±4.49%
When sampling from a known, finite population, you can apply the finite population correction:
Given:
Result: The required sample size is 323 (rounded up)
When sampling from a known, finite population, you can apply the finite population correction:
When the expected proportion is unknown, use p = 0.5 (50%) to calculate the most conservative (largest) sample size:
Example for 95% confidence level and 5% margin of error:
Result: The maximum required sample size is 385 (rounded up)
If a survey finds that 60% of respondents favor a policy with a margin of error of ±3% at the 95% confidence level:
"We are 95% confident that between 57% and 63% of the population favors the policy."
import numpy as np
from scipy import stats
def margin_of_error_proportion(p, n, confidence_level=0.95, population_size=None):
z = stats.norm.ppf((1 + confidence_level) / 2)
se = np.sqrt(p * (1 - p) / n)
moe = z * se
if population_size and population_size > n:
fpc = np.sqrt((population_size - n) / (population_size - 1))
moe *= fpc
return moe * 100
def margin_of_error_mean(std_dev, n, confidence_level=0.95, population_size=None):
z = stats.norm.ppf((1 + confidence_level) / 2)
se = std_dev / np.sqrt(n)
moe = z * se
if population_size and population_size > n:
fpc = np.sqrt((population_size - n) / (population_size - 1))
moe *= fpc
return moe
p = 0.7
n = 400
moe_prop = margin_of_error_proportion(p, n, 0.95)
print(f"Margin of Error for proportion: ±{moe_prop:.2f}%")
# Margin of Error for proportion: ±4.49%
std_dev = 15
n = 100
moe_mean = margin_of_error_mean(std_dev, n, 0.95)
print(f"Margin of Error for mean: ±{moe_mean:.2f}")
# Margin of Error for mean: ±2.94
#---------------------------------------------------#
def sample_size_for_proportion(p, desired_moe, confidence_level=0.95, population_size=None):
desired_moe_decimal = desired_moe / 100
z = stats.norm.ppf((1 + confidence_level) / 2)
n = (z**2 * p * (1-p)) / (desired_moe_decimal**2)
if population_size and population_size > 0:
n = (n * population_size) / (n + population_size - 1)
return int(np.ceil(n))
def sample_size_for_mean(std_dev, desired_moe, confidence_level=0.95, population_size=None):
z = stats.norm.ppf((1 + confidence_level) / 2)
n = (z**2 * std_dev**2) / (desired_moe**2)
if population_size is not None and population_size > 0:
n = (n * population_size) / (n + population_size - 1)
return int(np.ceil(n))
moe = 5
p = 0.5
n = sample_size_for_proportion(p, moe)
print(f"Sample size for proportion with {moe}% margin of error: {n}")
# Sample size for proportion with 5% margin of error: 385
std_dev = 15
moe = 3
n = sample_size_for_mean(std_dev, moe)
print(f"Sample size for mean with {moe} margin of error: {n}")
# Sample size for mean with 3 margin of error: 97