CONFIDENCE 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. Your sample mean, x, is at the center of this range and the range is x ± CONFIDENCE.

For example, if x is the sample mean of delivery times for products ordered through the mail, x ± CONFIDENCE is a range of population means. For any population mean, µ0, in this range, the probability of obtaining a sample mean further from µ0 than x is greater than alpha; for any population mean, µ0, not in this range, the probability of obtaining a sample mean further from µ0 than x is less than alpha. In other words, assume that we use x, standard_dev, and size to construct a two-tailed test at significance level alpha of the hypothesis that the population mean is µ0. Then we will not reject that hypothesis if µ0 is in the confidence interval and will reject that hypothesis if µ0 is not in the confidence interval. The confidence interval does not allow us to infer that there is probability 1 – alpha that our next package will take a delivery time that is in the confidence interval.

Important

This function was replaced with one or more new functions that may provide improved accuracy and whose names better reflect their usage. This function is still available for compatibility with earlier versions of Excel. However, if backward compatibility is not required, you should consider using the new functions from now on, because they more accurately describe their functionality.

Syntax

CONFIDENCE(alpha,standard_dev,size)

Argument

Description

Remarks

alpha

The significance level used to compute the confidence level. The confidence level equals 100*(1 - alpha)%, or in other words, an alpha of 0.05 indicates a 95 percent confidence level.

• If alpha ≤ 0 or alpha ≥ 1, CONFIDENCE returns the #NUM! error value.

• If this argument is nonnumeric, this function returns the #VALUE! error value.

standard_dev

The population standard deviation for the data range and is assumed to be known.

• If this argument is equal to or less than zero, this function returns the #NUM! error value.

• If this argument is nonnumeric, this function returns the #VALUE! error value.

size

The sample size.

• If size < 1, CONFIDENCE returns the #NUM! error value.

• If this argument is nonnumeric, this function returns the #VALUE! error value.

• If this argument contains a decimal value, this function ignores the numbers to the right side of the decimal point.

General remarks

If we assume alpha equals 0.05, we need to calculate the area under the standard normal curve that equals (1 - alpha), or 95 percent. This value is ± 1.96. The confidence interval is therefore:

Example

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. With alpha = .05, CONFIDENCE(.05, 2.5, 50) returns 0.69291. The corresponding confidence interval is then 30 ± 0.69291 = approximately [29.3, 30.7]. For any population mean, µ0, in this interval, the probability of obtaining a sample mean further from µ0 than 30 is more than 0.05. Likewise, for any population mean, µ0, outside this interval, the probability of obtaining a sample mean further from µ0 than 30 is less than 0.05.

To make the following example easier to understand, you can copy the data to a blank sheet and then enter the function underneath the data. Do not select the row or column headings (1, 2, 3... A, B, C...) when you copy the sample data to a blank sheet.

 1 2 3 4
 A B Data Description 0.05 Significance level 2.5 Standard deviation of the population 50 Sample size Formula Description (Result) =CONFIDENCE(A2,A3,A4) Confidence interval for a population mean. In other words, the confidence interval for the underlying population mean for travel to work equals 30 ± 0.692951 minutes, or 29.3 to 30.7 minutes. (0.692952)
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