Summary
This article describes the CONFIDENCE function in Microsoft Office Excel 2003 and in Microsoft Office Excel 2007, illustrates how the function is used, and compares the results of the function for Excel 2003 and for Excel 2007 with the results of CONFIDENCE in earlier versions of Excel.
The meaning of a confidence interval is frequently misinterpreted, and we try to provide an explanation of valid and invalid statements that can be made after you determine a CONFIDENCE value from your data.
More Information
The CONFIDENCE(alpha, sigma, n) function returns a value that you can use to construct a confidence interval for a population mean. The confidence interval is a range of values that are centered at a known sample mean. Observations in the sample are assumed to come from a normal distribution with known standard deviation, sigma, and the number of observations in the sample is n.
Syntax
CONFIDENCE(alpha,sigma,n)
Parameters: Alpha is a probability and 0 < alpha < 1. Sigma is a positive number, and n is a positive integer that corresponds to the sample size.
Typically, alpha is a small probability, such as 0.05.
Example of usage
Assume that intelligence quotient (IQ) scores follow a normal distribution with standard deviation 15. You test IQs for a sample of 50 students in your local school and obtain a sample mean of 105. You want to compute a 95% confidence interval for the population mean. A 95% or 0.95 confidence interval corresponds to alpha = 1 – 0.95 = 0.05.
To illustrate the CONFIDENCE function, create a blank Excel worksheet, copy the following table, and then select cell A1 in your blank Excel worksheet. On the Edit menu, click Paste.
Note: In Excel 2007, click Paste in the Clipboard group on the Home tab.
The entries in the table below fill cells A1:B7 in your worksheet.
|
alpha |
0.05 |
|
stdev |
15 |
|
n |
50 |
|
sample mean |
105 |
|
=CONFIDENCE(B1,B2,B3) |
|
|
=NORMSINV(1 - B1/2)*B2/SQRT(B3) |
After you paste this table into your new Excel worksheet, click the Paste Options button, and then click Match Destination Formatting.
With the pasted range still selected, point to Column on the Format menu, and then click AutoFit Selection.
Note: In Excel 2007, with the pasted range of cells selected, click Format in the Cells group on the Home tab, and then click AutoFit Column Width.
Cell A6 shows the value of CONFIDENCE. Cell A7 shows the same value because a call to CONFIDENCE(alpha, sigma, n) returns the result of computing:
NORMSINV(1 – alpha/2) * sigma / SQRT(n)
No changes were made directly to CONFIDENCE, but NORMSINV was improved in Microsoft Excel 2002, and then more improvements were made between Excel 2002 and Excel 2007. Therefore, CONFIDENCE may return different (and improved) results in these later versions of Excel, because CONFIDENCE relies on NORMSINV.
This does not mean that you should lose confidence in CONFIDENCE for earlier versions of Excel. Inaccuracies in NORMSINV generally occurred for values of its argument very close to 0 or very close to 1. In practice, alpha is generally set to 0.05, 0.01, or maybe 0.001. Values of alpha have to be much smaller than that, for example 0.0000001, before round-off errors in NORMSINV are likely to be noticed.
Note: See the article on NORMSINV for a discussion of computational differences in NORMSINV.
For more information, click the following article number to view the article in the Microsoft Knowledge Base:
826772 Excel statistical functions: NORMSINV
Interpretation of the results of CONFIDENCE
The Excel Help file for CONFIDENCE has been rewritten for Excel 2003 and for Excel 2007 because all earlier versions of the Help file gave misleading advice on interpreting results. The example states, "Suppose we observe that, in our sample of 50 commuters, the average length of travel to work is 30 minutes with a population standard deviation of 2.5. We can be 95 percent confident that the population mean is in the interval 30 +/- 0.692951" where 0.692951 is the value returned by CONFIDENCE(0.05, 2.5, 50).
For the same example, the conclusion reads, "the average length of travel to work equals 30 ± 0.692951 minutes, or 29.3 to 30.7 minutes." Presumably, this is also a statement about the population mean falling within the interval [30 – 0.692951, 30 + 0.692951] with probability 0.95.
Before conducting the experiment that yielded the data for this example, a classical statistician (as opposed to a Bayesian statistician) can make no statement about the probability distribution of the population mean. Instead, a classical statistician deals with hypothesis testing.
For example, a classical statistician may want to conduct a two-sided hypothesis test that is based on the supposition of a normal distribution with known standard deviation (such as 2.5), a particular pre-selected value of the population mean, µ0, and a pre-selected significance level (such as 0.05). The test's result would be based on the value of the observed sample mean (for example 30) and the null hypothesis that the population mean is µ0 would be rejected at a significance level 0.05 if the observed sample mean was too far from µ0 in either direction. If the null hypothesis is rejected, the interpretation is that a sample mean that far or further from µ0 would occur by chance less than 5% of the time under the supposition that µ0 is the true population mean. After conducting this test, a classical statistician still cannot make any statement about the probability distribution of the population mean.
A Bayesian statistician, on the other hand, would start with an assumed probability distribution for the population mean (named an a priori distribution), would gather experimental evidence in the same way as the classical statistician, and would use this evidence to revise her or his probability distribution for the population mean and thereby obtain an a posteriori distribution. Excel provides no statistical functions that would help a Bayesian statistician in this endeavor. Excel's statistical functions are all intended for classical statisticians.
Confidence intervals are related to Hypothesis Tests. Given the experimental evidence, a confidence interval makes a concise statement about the values of the hypothesized population mean µ0 that would yield acceptance of the null hypothesis that the population mean is µ0 and the values of µ0 that would yield rejection of the null hypothesis that the population mean is µ0. A classical statistician cannot make any statement about the chance that the population mean falls in any specific interval, because she or he never makes a priori assumptions about this probability distribution and such assumptions would be required if one were to use experimental evidence to revise them.
Explore the relationship between hypothesis tests and confidence intervals by using the example at the beginning of this section. With the relationship between CONFIDENCE and NORMSINV stated in the last section, you have:
CONFIDENCE(0.05, 2.5, 50) = NORMSINV(1 – 0.05/2) * 2.5 / SQRT(50) = 0.692951
Because the sample mean is 30, the confidence interval is 30 +/- 0.692951.
Now consider a two-sided hypothesis test with the significance level 0.05 as described earlier that assumes a normal distribution with standard deviation 2.5, a sample size of 50 and a specific hypothesized population mean, µ0. If this is the true population mean, then the sample mean will come from a normal distribution with population mean µ0 and standard deviation, 2.5/SQRT(50). This distribution is symmetrical about µ0 and you would want to reject the null hypothesis if ABS(sample mean - µ0) > some cutoff value. The cutoff value would be such that if µ0 were the true population mean, a value of sample mean - µ0 higher than this cutoff or a value of µ0 – sample mean higher than this cutoff would each occur with probability 0.05/2. This cutoff value is
NORMSINV(1 – 0.05/2) * 2.5/SQRT(50) = CONFIDENCE(0.05, 2.5, 50) = 0. 692951
So reject the null hypothesis (population mean = µ0) if one of the following statements is true:
sample mean - µ0 > 0. 692951
0 – sample mean > 0. 692951
Because sample mean = 30 in our example, these two statements become the following statements:
30 - µ0 > 0. 692951
µ0 – 30 > 0. 692951
Rewriting them so that only µ0 appears on the left yields the following statements:
µ0 < 30 - 0. 692951
µ0 > 30 + 0. 692951
These are exactly the values of µ0 that are not in the confidence interval [30 – 0.692951, 30 + 0.692951]. Therefore, the confidence interval [30 – 0.692951, 30 + 0.692951] contains those values of µ0 where the null hypothesis that the population mean is µ0 would not be rejected, given the sample evidence. For values of µ0 outside this interval, the null hypothesis that the population mean is µ0 would be rejected given the sample evidence.
Conclusions
Inaccuracies in earlier versions of Excel generally occur for extremely small or extremely large values of p in NORMSINV(p). CONFIDENCE is evaluated by calling NORMSINV(p), so accuracy of NORMSINV is a potential concern for users of CONFIDENCE. However, values of p that are used in practice are not likely to be extreme enough to cause significant round-off errors in NORMSINV, and performance of CONFIDENCE should not be a concern to users of any version of Excel.
Most of this article has focused on interpreting the results of CONFIDENCE. In other words, we have asked, "What is the meaning of a confidence interval?" Confidence intervals are frequently misunderstood. Unfortunately, Excel Help files in all versions of Excel that are earlier than Excel 2003 have contributed to this misunderstanding. The Excel 2003 Help file has been improved.
