DEVSQ

Returns the sum of squares of deviations from the mean.

Syntax:

DEVSQ(number1, number2, ... number30)

number1 to number30 are up to 30 numbers or ranges containing numbers.

DEVSQ calculates the mean of all the numbers, then sums the squared deviation of each number from that mean. With N values, the calculation formula is:

$\sum_{i = 1}^{N} (x_{i} - \overline{x})^{2}$

Example:

DEVSQ(1, 3, 5)

returns 8, calculated as $(- 2)^{2} + 0 + (2)^{2}$

Application:

Analyzing Daily Temperature Fluctuations

The DEVSQ function is a statistical tool used to calculate the sum of the squared deviations from the mean. It's a key component in calculating variance and standard deviation, which are measures of how spread out a set of data is. A high DEVSQ value indicates that the data points are widely dispersed from the average, while a low DEVSQ value indicates that the data points are clustered closely around the average.

Let's imagine you are a meteorologist and you want to analyze the consistency of the daily high temperatures in your city over a week. A lower DEVSQ value for the temperature data would suggest that the weather has been very stable, while a higher value would indicate a week of unpredictable, fluctuating temperatures.

Here is a table showing the daily high temperatures (in Fahrenheit) for one week:

	Day	High Temperature ( $x_{i}$ )	Mean ( $\overset{x}{ˉ}$ )	Deviation ( $x_{i} - \overset{x}{ˉ}$ )	Squared Deviation ( $(x_{i} - \overset{x}{ˉ})^{2}$ )
	A	B	C	D	E
1	Monday	75	77.14	-2.14	4.58
2	Tuesday	78	77.14	0.86	0.74
3	Wednesday	79	77.14	1.86	3.46
4	Thursday	76	77.14	-1.14	1.3
5	Friday	80	77.14	2.86	8.18
6	Saturday	74	77.14	-3.14	9.86
7	Sunday	78	77.14	0.86	0.74
8	Sum	540			28.86

Steps to calculate DEVSQ:

Find the Mean ( $\overset{x}{ˉ}$ ): Sum all the daily temperatures and divide by the number of days.
- Sum = 75 + 78 + 79 + 76 + 80 + 74 + 78 = 540
- Number of days = 7
- Mean ( $\overset{x}{ˉ}$ ) = 540 / 7 ≈ 77.14
Find the Deviation ( $x_{i} - \overset{x}{ˉ}$ ): Subtract the mean from each daily temperature.
- Monday: 75 - 77.14 = -2.14
- Tuesday: 78 - 77.14 = 0.86
- ...and so on.
Find the Squared Deviation ( $(x_{i} - \overset{x}{ˉ})^{2}$ ): Square each of the deviations. This step is crucial because it ensures all values are positive and gives more weight to larger deviations, which is a key part of statistical analysis.
- Monday: (−2.14)2 ≈ 4.58
- Tuesday: (0.86)2 ≈ 0.74
- ...and so on.
Find the Sum of the Squared Deviations (DEVSQ): Add all the squared deviations together. This is the value the DEVSQ function returns.
- DEVSQ = 4.58 + 0.74 + 3.46 + 1.30 + 8.18 + 9.86 + 0.74 = 28.86

Interpretation:

The DEVSQ value of approximately 28.86 gives you a single number that quantifies the overall variability of the temperatures during that week. You could use this value to compare this week's weather stability to previous weeks. For instance, if the DEVSQ for the following week was 15, you would know that the temperatures were more consistent and stable during that second week.

Result for DEVSQ:

28.86