Statistics is all about drawing conclusions in the face of uncertainty. Whenever you take a sample, you can’t be completely certain that your sample truly reflects the population it’s drawn from. Statisticians deal with this uncertainty by taking the factors that could impact the estimate into account, quantifying their uncertainty and performing statistical tests to draw conclusions from this uncertain data.
Statisticians use confidence intervals to specify a range of values that is likely to contain the “true” population mean on the basis of a sample, and express their level of certainty in this through confidence levels. While calculating confidence levels isn’t often useful, calculating confidence intervals for a given confidence level is a very useful skill.
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Calculate a confidence interval for a given confidence level by multiplying the standard error by the Z score for your chosen confidence level. Subtract this result from your sample mean to get the lower bound, and add it to the sample mean to find the upper bound. (See Resources)
Repeat the same process but with the t score in place of the Z score for smaller samples (n < 30).
Find a confidence level for a data set by taking half of the size of the confidence interval, multiplying it by the square root of the sample size and then dividing by the sample standard deviation. Look up the resulting Z or t score in a table to find the level.
The Difference between Confidence Level vs. Confidence Interval
When you see a statistic quoted, there is sometimes a range given after it, with the abbreviation “CI” (for “confidence interval”) or simply a plus-minus symbol followed by a figure. For instance, “the mean weight of an adult male is 180 pounds (CI: 178.14 to 181.86)” or “the mean weight of an adult male is 180 ± 1.86 pounds.” These both tell you the same information: based on the sample used, the mean weight of a man probably falls within a certain range. The range itself is called the confidence interval.
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If you want to be as sure as possible that the range contains the true value, then you can widen the range. This would increase your “confidence level” in the estimate, but the range would cover more potential weights. Most statistics (including the one quoted above) are given as 95 percent confidence intervals, which mean that there is a 95 percent chance that the true mean value is within the range. You can also use a 99 percent confidence level or a 90 percent confidence level, depending on your needs.
Calculating Confidence Intervals or Levels for Large Samples
When you use a confidence level in statistics, you usually need it to calculate a confidence interval. This is a bit easier to do if you have a large sample, for example, over 30 people, because you can use Z score for your estimate rather than more complicated t scores.
Take your raw data and calculate the sample mean (simply add up the individual results and divide by the number of results). Calculate the standard deviation by subtracting the mean from each individual result to find the difference and then square this difference. Add up all of these differences and then divide the result by the sample size minus 1. Take the square root of this result to find the sample standard deviation (See Resources).
Determine the confidence interval by first finding the standard error:
SE = s / √n
Where s is your sample standard deviation and n is your sample size. For example, if you took a sample of 1,000 men to figure the average weight of a man, and got a sample standard deviation of 30, this would give:
SE = 30 / √1000 = 30 / 31.62 = 0.95
To find the confidence interval from this, look up the confidence level you want to calculate the interval for in a Z-score table and multiply this value by the Z score. For a 95 percent confidence level, the Z-score is 1.96. Using the example, this means:
Mean ± Z × SE= 180 pounds ± 1.96 × 0.95 = 180 ± 1.86 pounds
Here, ± 1.86 pounds is the 95 percent confidence interval.
If you have this bit of information instead, along with the sample size and the standard deviation, you can calculate the confidence level by using the following formula:
Z = 0.5 × size of confidence interval × √n / s
The size of the confidence interval is just twice the ± value, so in the example above, we know 0.5 times this is 1.86. This gives:
Z = 1.86 × √1000 / 30 = 1.96
This gives us a value for Z, which you can look up in a Z-score table to find the corresponding confidence level.
Calculating Confidence Intervals for Small Samples
For small samples, there is a similar process for calculating the confidence interval. First, subtract 1 from your sample size to find your “degrees of freedom.” In symbols:
df = n −1
For a sample n = 10, this gives df = 9.
Find your alpha value by subtracting the decimal version of the confidence level (i.e. your percentage confidence level divided by 100) from 1 and dividing the result by 2, or in symbols:
α = (1 – decimal confidence level) / 2
So for a 95 percent (0.95) confidence level:
α = (1 – 0.95) / 2 = 0.05 / 2 = 0.025
Look up your alpha value and degrees of freedom in a (one tail) t distribution table and make note of the result. Alternatively, omit the division by 2 above and use a two-tail t value. In this example, the result is 2.262.
As in the previous step, calculate the confidence interval by multiplying this number by the standard error, which is determined using your sample standard deviation and sample size in the same way. The only difference is that in place of the Z score, you use the t score.