When we deal with percentages derived from sample data, understanding the potential error or variability is essential. A common method to gauge this variability is by calculating the standard deviation of the sampling distribution of a proportion.
Formula:
Standard Deviation (SD) of proportion, p:
SD = Square Root [(p * (1 - p)) / n]
Where:
SD: Standard deviation of the proportion’s sampling distribution.p: Sample proportion (expressed as a decimal, so 70% would be 0.7).n: Sample size.
Example:
Consider a political survey in a town regarding support for a particular candidate. From a random sample of 400 residents, 280 express their support. The sample proportion, p, is 280/400 = 0.7 (70%).
Using our formula:SD = Square Root [(0.7 * (1 - 0.7)) / 400]SD ≈ 0.023 or 2.3%
This suggests that in our sample, the standard deviation of the residents’ support proportion for the candidate is about 2.3%.
Deep Dive:
Our formula assumes the proportion’s sampling distribution is roughly normal. This is generally true when both n * p and n * (1 - p) are greater than 5.
To compute a confidence interval for this proportion, the standard deviation isn’t enough. We also need the z-value, from the z-distribution, for our desired confidence level. The margin of error is:
Margin of Error = z * SD
For a 95% confidence level, we often use a z-value of 1.96.
If comparing proportions from two independent samples, the formula gets more involved. In such cases, the two-proportion z-test is used to see if there’s a significant difference.
In conclusion, this formula gives insight into the variability in proportion measurements. It’s crucial to apply it correctly, keeping in mind its assumptions and limitations.
