How to use the CONFIDENCE.T function
What is the CONFIDENCE.T function?
The CONFIDENCE.T function calculates the confidence range for a population mean using a Student's t distribution. Use the CONFIDENCE.T function when you sample size is small (less than 30 observations). The t-distribution accounts for the increased uncertainty associated with small sample sizes and provides a more accurate estimate of the confidence interval.
The CONFIDENCE.NORM function is recommended if the sample size is larger than 30 observations.
Table of Contents
1. Introduction
What is the confidence range for a population mean?
A confidence interval around a sample mean provides a range of plausible values for the unknown population mean. It is calculated using the sample mean, standard error, and t-distribution, indicating the precision of the sample mean as an estimate of the actual population mean.
What is a Student's t distribution?
The Student's t distribution is a probability distribution that arises when estimating population means from samples, depending on the sample size and used to construct confidence intervals and in Student's t-tests.
The T distribution has heavier tails that accounts for the increased uncertainty associated with small sample sizes. The larger sample size the thinner tails, as size goes to infinity the t distribution becomes the normal distribution.
The image above shows the t distribution with one degree of freedom, the tails are very heavy. The normal distribution, which is also shown in the image above, has very thin tails.
2. Syntax
CONFIDENCE.T(alpha,standard_dev,size)
3. Arguments
| alpha | Required. The significance level. |
| standard_dev | Required. The standard deviation. |
| size | Required. The number of samples. |
What is the significance level?
The significance level, denoted α (alpha) , is the probability of that you want to use to calculate the confidence level. For example, a 95% significance level corresponds to a 0.05 alpha. 1 - 0.95 = 0.05
What is standard deviation?
Standard deviation tells you how far from the average values are spread out. Both charts above have numbers and an average plotted, they share the exact same average however, the numbers are not the same.
Chart A above shows that the values are more spread out than the values in chart B. Chart A has a standard deviation of 23.45256334, standard deviation for chart B is 5.207075606.
Standard deviation measures the amount of variation or dispersion of values in a dataset relative to the mean. It is calculated as the square root of the variance by taking
- each data point's deviation from the mean
- squaring it
- averaging the squares
- finally square rooting the result
Standard deviation = √(Σ(x - x̄)/n)
x̄ - arithmetic mean
n - count of values
4. Example 1
A marketing firm conducted a survey to estimate the average monthly expenditure of households in a city. They surveyed 20 randomly selected households and recorded their monthly expenditures. Calculate the 99% confidence interval for the mean monthly household expenditure in the city.
Average monthly expenditure : 1000
Standard deviation: 100
Size: 20
The CONFIDENCE.T function is what we want to use since the sample size is only 20. It lets you calculate the interval that satisfies a given a significance level, standard deviation and a number representing the population size based on the t distribution.
The image above shows the arguments:
- alpha in cell C19 which is 1-0.99 = 0.01
- standard_dev in cell C20 = 100
- size in cell C21 = 20
Formula in cell C23:
The CONFIDENCE.T function returns approx 63.97 for the specified arguments above.We can now calculate the upper and lower limit by adding the confidence interval from cell C23 to the mean μ. I have done this in cell C25:
The lower limit is calculated by subtracting the confidence level to the mean μ. Cell 26:
In other words, there is a 99% probability that the average monthly expenditure is between approx. 936.03 and approx. 1063.97
5. Example 2
A nutritionist wants to estimate the average daily calorie intake of male adults in a certain region. They collect data on the daily calorie intake of 10 randomly selected male adults. Using this sample data in cell range E19:H21, calculate the 95% confidence interval for the mean daily calorie intake of male adults in the region. Here are the observations:
| 2194 | 2649 | 2407 | 2123 |
| 2896 | 2788 | 2064 | 2916 |
| 2249 | 2421 |
The CONFIDENCE.NORM function requires three arguments, the alpha, standard deviation and the size. The alpha is specified in question, 1 - 0.95 = 0.05
We can calculate the mean μ using the AVERAGE function. Formula in cell C24:
The average is calculated by adding all the measurements and then divide by the number of observations. The standard deviation is also easy to calculate in Excel. Formula in cell C25:
We now have all the arguments needed to calculate the interval that satisfies a given significance level at 80% (0.8).
- alpha in cell C19 which is 1-0.8 = 0.05
- standard_dev in cell C20 approx. equal to 321.48
- size in cell C21 = 10
Formula in cell C23:
The CONFIDENCE.T function returns 229.98 for the specified arguments above. We can now calculate the upper and lower limit by adding the confidence interval from cell C23 to the mean μ. I have done this in cell C25:
The lower limit is calculated by subtracting the confidence level to the mean μ. Cell 26:
In other words, there is a 95% probability that the mean daily calorie intake is between approx. 2240 and approx. 2701
6. Example 3
A study aims to determine the average reaction time of drivers when presented with a sudden obstacle on the road. The researchers recorded the reaction times of 15 randomly selected drivers in a controlled experiment. Calculate the 90% confidence interval for the mean reaction time of drivers in this population.
Average reaction time : 459 ms
Standard deviation: 78 ms
Size: 15
alpha: 1-0.9 = 0.1
Formula in cell C23:
The CONFIDENCE.T function returns approx 35.47 for the specified arguments above.We can now calculate the upper and lower limit by adding the confidence interval from cell C23 to the mean μ. I have done this in cell C25:
The lower limit is calculated by subtracting the confidence level to the mean μ. Cell 26:
In other words, there is a 90% probability that the average reaction time is between approx. 423.53 and approx. 494.47
7. Function not working
The CONFIDENCE.T function returns:
- #VALUE! error value if any argument is non-numeric.
- #NUM! error value if
- alpha <= 0 (zero)
- alpha >= 1.
- standard_dev <= 0.
- size < 1.
- #DIV/0! error value if size equals 1.
CONFIDENCE.T function truncates size argument to integers.
7.1 Troubleshooting the error value
When you encounter an error value in a cell a warning symbol appears, displayed in the image above. Press with mouse on it to see a pop-up menu that lets you get more information about the error.
- The first line describes the error if you press with left mouse button on it.
- The second line opens a pane that explains the error in greater detail.
- The third line takes you to the "Evaluate Formula" tool, a dialog box appears allowing you to examine the formula in greater detail.
- This line lets you ignore the error value meaning the warning icon disappears, however, the error is still in the cell.
- The fifth line lets you edit the formula in the Formula bar.
- The sixth line opens the Excel settings so you can adjust the Error Checking Options.
Here are a few of the most common Excel errors you may encounter.
#NULL error - This error occurs most often if you by mistake use a space character in a formula where it shouldn't be. Excel interprets a space character as an intersection operator. If the ranges don't intersect an #NULL error is returned. The #NULL! error occurs when a formula attempts to calculate the intersection of two ranges that do not actually intersect. This can happen when the wrong range operator is used in the formula, or when the intersection operator (represented by a space character) is used between two ranges that do not overlap. To fix this error double check that the ranges referenced in the formula that use the intersection operator actually have cells in common.
#SPILL error - The #SPILL! error occurs only in version Excel 365 and is caused by a dynamic array being to large, meaning there are cells below and/or to the right that are not empty. This prevents the dynamic array formula expanding into new empty cells.
#DIV/0 error - This error happens if you try to divide a number by 0 (zero) or a value that equates to zero which is not possible mathematically.
#VALUE error - The #VALUE error occurs when a formula has a value that is of the wrong data type. Such as text where a number is expected or when dates are evaluated as text.
#REF error - The #REF error happens when a cell reference is invalid. This can happen if a cell is deleted that is referenced by a formula.
#NAME error - The #NAME error happens if you misspelled a function or a named range.
#NUM error - The #NUM error shows up when you try to use invalid numeric values in formulas, like square root of a negative number.
#N/A error - The #N/A error happens when a value is not available for a formula or found in a given cell range, for example in the VLOOKUP or MATCH functions.
#GETTING_DATA error - The #GETTING_DATA error shows while external sources are loading, this can indicate a delay in fetching the data or that the external source is unavailable right now.
7.2 The formula returns an unexpected value
To understand why a formula returns an unexpected value we need to examine the calculations steps in detail. Luckily, Excel has a tool that is really handy in these situations. Here is how to troubleshoot a formula:
- Select the cell containing the formula you want to examine in detail.
- Go to tab “Formulas” on the ribbon.
- Press with left mouse button on "Evaluate Formula" button. A dialog box appears.
The formula appears in a white field inside the dialog box. Underlined expressions are calculations being processed in the next step. The italicized expression is the most recent result. The buttons at the bottom of the dialog box allows you to evaluate the formula in smaller calculations which you control. - Press with left mouse button on the "Evaluate" button located at the bottom of the dialog box to process the underlined expression.
- Repeat pressing the "Evaluate" button until you have seen all calculations step by step. This allows you to examine the formula in greater detail and hopefully find the culprit.
- Press "Close" button to dismiss the dialog box.
There is also another way to debug formulas using the function key F9. F9 is especially useful if you have a feeling that a specific part of the formula is the issue, this makes it faster than the "Evaluate Formula" tool since you don't need to go through all calculations to find the issue..
- Enter Edit mode: Double-press with left mouse button on the cell or press F2 to enter Edit mode for the formula.
- Select part of the formula: Highlight the specific part of the formula you want to evaluate. You can select and evaluate any part of the formula that could work as a standalone formula.
- Press F9: This will calculate and display the result of just that selected portion.
- Evaluate step-by-step: You can select and evaluate different parts of the formula to see intermediate results.
- Check for errors: This allows you to pinpoint which part of a complex formula may be causing an error.
The image above shows cell reference C3 converted to hard-coded value using the F9 key. The CONFIDENCE.T function requires numerical values which is not the case in this example. We have found what is wrong with the formula.
Tips!
- View actual values: Selecting a cell reference and pressing F9 will show the actual values in those cells.
- Exit safely: Press Esc to exit Edit mode without changing the formula. Don't press Enter, as that would replace the formula part with the calculated value.
- Full recalculation: Pressing F9 outside of Edit mode will recalculate all formulas in the workbook.
Remember to be careful not to accidentally overwrite parts of your formula when using F9. Always exit with Esc rather than Enter to preserve the original formula. However, if you make a mistake overwriting the formula it is not the end of the world. You can “undo” the action by pressing keyboard shortcut keys CTRL + z or pressing the “Undo” button
7.3 Other errors
Floating-point arithmetic may give inaccurate results in Excel - Article
Floating-point errors are usually very small, often beyond the 15th decimal place, and in most cases don't affect calculations significantly.
Functions in 'Statistical' category
The CONFIDENCE.T function function is one of 73 functions in the 'Statistical' category.
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