FTEST

Returns the result of an F-test.

Syntax:

FTEST(data1, data2)

data1 and data2 are ranges or arrays (possibly of different size) containing numbers, on which the F-test is performed. The F-test calculates the likelihood that two samples have the same variance.

In effect, the sample variances of data1 and data2 are calculated. data1 and data2 are re-ordered if necessary so that data1 has the larger variance (σ1) and data2 the smaller (σ2), and an F_value is calculated as σ1/σ2. The result returned by FTEST is 2*FDIST(F_value; COUNT(data1)-1; COUNT(data2)-1). This is the two-tailed probability that the variances in data1 and data2 are not significantly different. (The internal algorithms used by Calc are more sophisticated than this).

Example:

FTEST({9,8,6,8}, {5,6,7})

returns approximately 0.82. The probability that the variances of these two samples are not significantly different is about 82%. Note: a real world example would need more data than this.

Application:

Let's imagine a scenario where a manager wants to compare the consistency of two different production lines that produce the same type of product. The goal is to see if one production line has a more consistent output (i.e., less variance) than the other.

The Hypothesis

Null Hypothesis ( $H_{0}$ ): The variances of the two production lines are equal. ( $σ_{1}^{2} = σ_{2}^{2}$ )
Alternative Hypothesis ( $H_{a}$ ): The variances of the two production lines are not equal. ( $σ_{1}^{2} \neq = σ_{2}^{2}$ )

The Data

The manager collects a random sample of 10 products from Production Line A and 12 products from Production Line B. The weight of each product (in grams) is recorded.

Table 1: Sample Data

	Production Line A (grams)	Production Line B (grams)
	A	B
1	102.5	100.8
2	98.7	103.1
3	100.4	99.5
4	101.9	101.5
5	99.2	104.2
6	103.1	98.9
7	97.5	102.7
8	104	97.1
9	96.8	100.3
10	105.3	103.5
11		99.8
12		101.2

Calculation

To perform the F-test, we first need to calculate the variance for each data set.

Sample Variance for Production Line A ( $s_{1}^{2}$ ): 6.51
Sample Variance for Production Line B ( $s_{2}^{2}$ ): 4.67

The F-statistic is the ratio of the larger sample variance to the smaller sample variance.

$F = \frac{L a r g er Va r ian ce}{S ma ll er Va r ian ce} = \frac{6.51}{4.67} = 1.39$

Using the F.TEST Function

You could use the F.TEST function to get the p-value directly. The syntax is F.TEST(array1, array2).

array1: The range of data for Production Line A (102.5, 98.7, etc.)
array2: The range of data for Production Line B (100.8, 103.1, etc.)

The function would return the two-tailed probability that the variances are not significantly different. A typical result for this data might be around 0.32.

Conclusion

With a p-value of 0.32, which is much greater than the common significance level of 0.05, we fail to reject the null hypothesis. This means there is not enough statistical evidence to conclude that the variances of the two production lines are significantly different. Based on this test, the manager can conclude that both production lines are similarly consistent in their output.

Result for F.TEST(array1, array2):

0.32