How to use the ABS function

What is the ABS function?

The ABS function converts negative numbers to positive numbers, in other words, the ABS function removes the sign and returns the absolute value.

1. Introduction

When is it useful to calculate the absolute value?

The absolute value in math is expressed like this: |-12| = 12 The absolute value is denoted by vertical bars on each side of the number. The ABS function lets you calculate the non-negative value.

Here are a few examples when calculating the absolute values are useful:

• The average of absolute deviations from the arithmetic mean, here is the formula:
  Average deviation from the arithmetic mean = (1/n) * Σ(i=1 to n) |x_i - m(X)|
  n - number of values
  x_i - each value
  m(X) - arithmetic mean
• The average of the absolute deviations from the median, here is the formula:
  Average deviation from the median= (1/n) * Σ(i=1 to n) |x_i - m(X)|
  n - number of values
  x_i - each value
  m(X) - median
• The average of the absolute deviations from the mode, here is the formula:
  Average deviation from the mode = (1/n) * Σ(i=1 to n) |x_i - m(X)|
  n - number of values
  x_i - each value
  m(X) - mode

What is the arithmetic mean? It is the average of a data set.

What is the median? The median is the middle number of a group of numbers.

What is the mode? The most frequent number in a data set.

1. ABS Function Syntax

ABS(number)

number

The number you want to remove the sign from.

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2. Example

Calculate the absolute values based on this data set: [56, 24, -20, -23, 52, -70]?

The argument is:

• number - B3 (This is the cell reference to the actual number you want to calculate the absolute value of)

Formula in cell C3:

The number in cell B3 is 56, the ABS function returns 56. The remaining values in cells below B3 are also converted to their absolute values if you copy cell B3 and paste to cells below as far as needed.

The result from cell range B3:B8 is [56, 24, 20, 23, 52, 70]. They are all positive numbers meaning no negative numbers are present.

The ABS function converts a negative number to a positive number, in other words, it removes the sign from a number. Nothing happens to positive numbers.

The ABS function returns #VALUE if you use a text string, however, it returns 1 for boolean value TRUE and 0 (zero) for FALSE.

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4. How to convert a negative number to positive numbers and vice versa?

Change the signs based on this data set: [56, 24, -20, -23, 52, -70]?

Formula in cell C3:

The formula in cell C3 performs a sign reversal operation where negative numbers are converted to positive and positive numbers are converted to negative. This is achieved by multiplying the value in the cell reference B3, which in this example is 56, by -1.

The asterisk * symbol in the formula represents multiplication allowing you to multiply values in an Excel formula. While the minus sign (-) typically indicates subtraction, in this particular case, it acts as a sign operator because it immediately follows the multiplication operator *.

Interestingly, when you multiply a negative number by a negative number, the two minus signs effectively cancel each other out, resulting in a positive number. This behavior is the key to the sign reversal operation performed by the formula in cell C3.

The result of the sign reversal operation is [-56, -24, 20, 23, -52, 70]

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4.1 Explaining formula

Step 1 - Populate cell reference

B3*-1

returns

56*-1

Step 2 - Evaluate formula

56*-1 equals -56.

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5. How to convert all numbers to negative numbers?

Change the sign for all positive numbers in this data set: [56, 24, -20, -23, 52, -70]?

Formula in cell C3:

The formula in cell C3 converts all numbers to negative numbers. ABS(B3) calculates the absolute value of the number in cell B3. The ABS function returns the positive value of a number, regardless of its original sign. For example, B3 contains 56, ABS(56) returns 56.

The result of `ABS(B3)` is then multiplied by -1 by using the asterisk *-1. For example, 56*-1 returns -56.

When a positive number is multiplied by -1, it becomes negative. For example, 56* -1 = -56.
When a negative number is multiplied by -1, it becomes positive. For example, -56* -1 = -56. However, the ABS function makes sure no negative numbers are evaluated.

Therefore, the overall effect of the formula =ABS(B3)*-1 is to make all numbers negative regardless of the original sign.The result is [-56, -24, -20, -23, -52, -70]

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Explaining formula

Step 1 - Calculate the absolute number

ABS(B3)

becomes

ABS(56)

and returns 56.

Step 2 - Multiple with -1

The asterisk lets you multiply numbers in an Excel formula.

ABS(B3)*-1

becomes

56*-1

and returns -56.

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6. Convert negative numbers to positive and calculate sum

Calculate the total of this data set: [56, 24, -20, -23, 52, -70] However, use only absolute numbers?

ABS function arguments:

B3:B8 - A cell reference to the numbers you want to use. In this case, this is the specified data set described in the question above.

Formula in cell D3:

The formula converts negative numbers to positive and then sums the numbers. 56 + 24 + 20 + 23 + 52 + 70 equals 245. Excel 365 subscribers can use the SUM function instead of the SUMPRODUCT function since array formulas are performed in the same way as regular formulas in Excel 365.

The SUMPRODUCT function multiplies all values by 1 and then calculates a total. The SUMPRODUCT function does not require you to enter the formula as an array formula in earlier Excel versions than Excel 365, that is why it is used instead of the SUM function.

Explaining formula

Excel 365 users can use the following smaller formula:

Step 1 - Convert to absolute numbers

ABS(B3:B8)

becomes

ABS({56; 24; -20; -23; 52; -70})

and returns

{56; 24; 20; 23; 52; 70}.

Step 2 - Add absolute numbers and return a total

The SUM function adds numbers and then returns a total.

SUM(ABS(B3:B8))

becomes

SUM({56; 24; 20; 23; 52; 70})

and returns 245.

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7. Convert negative numbers to positive and calculate average

Calculate the average of the absolute numbers in this data set: [56, 24, -20, -23, 52, -70]?

ABS function arguments:

B3:B8 - A cell reference to the numbers you want to use. In this case, this is the specified data set described in the question above.

Array formula in cell D3:

The formula in cell D3 is an array formula, however, Excel 365 users can simply enter the formula as a regular formula. The formula removes the negative signs from the numbers in cell range B3:B8 and then calculates an average.

Here is the formula to calculate the arithmetic mean of absolute values:
Average deviation from the arithmetic mean = (1/n) * Σ(i=1 to n) | x_i |

• n - number of values
• x_i - each value

The absolute values are [56, 24, 20, 23, 52, 70]. The total is 56 + 24 + 20 + 23 + 52 + 70 equals 245. To calculate the average we need to divide by the number of values which in this example is 6. 245 / 6 equals 40.8333333333333

This value matches the calculated value in cell D3.

7.1 How to enter an array formula

1. Copy the array formula above.
2. Double press with the left mouse button on cell D3, a prompt appears.
3. Paste it to cell C3, shortcut keys are CTRL + v.
4. Press and hold CTRL + SHIFT keys simultaneously.
5. Press Enter once.
6. Release all keys.

The formula bar shows a beginning and ending curly bracket, don't enter these characters yourself.

7.2 Explaining formula

Step 1 - Convert numbers to absolute numbers

ABS(B3:B8)

becomes

ABS({56; 24; -20; -23; 52; -70})

and returns

{56; 24; 20; 23; 52; 70}.

Step 2 - Calculate an average

The AVERAGE function calculates the average of numbers in a cell range.

AVERAGE(number1, [number2], ...)

AVERAGE (ABS(B3:B8))

becomes

AVERAGE ({56; 24; 20; 23; 52; 70})

and returns approx. 40.83333

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8. Convert negative numbers to positive and calculate median

Calculate the median based on absolute numbers in this data set: [56, 24, -20, -23, 52, -70]?

ABS function arguments:

B3:B8 - A cell reference to the numbers you want to use. In this case, this is the specified data set described in the question above.

Array formula in cell D3:

How to enter an array formula. Excel 365 users can ignore the instructions for the array formula creation, it is not needed in Excel 365, simply press Enter after typing the formula.

The median is the middle number of a group of numbers. In this example the group of numbers consists of the absolute numbers in [56, 24, -20, -23, 52, -70]

They are [56, 24, 20, 23, 52, 70] and if we sort them from small to large we get: 20, 23, 24,52, 56, and 70. The middle numbers are 24 and 52, the average of 24 and 52 is 24 + 52 equals 76. 76 / 2 equals 38. The median is 38 which matches the value in cell D3.

Explaining array formula

Step 1 - Convert to absolute numbers

ABS(B3:B8)

becomes

ABS({56; 24; -20; -23; 52; -70})

and returns

{56; 24; 20; 23; 52; 70}.

Step 2 - Calculate median

The MEDIAN function calculates the median based on a group of numbers. The median is the middle number of a group of numbers.

If the group contains an even number of numbers the MEDIAN function calculates the average of the two numbers in the middle.

MEDIAN(ABS(B3:B8))

becomes

MEDIAN({56; 24; 20; 23; 52; 70})

and returns 38. 52 + 24 equals 76. 76 / 2 equals 38.

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Useful resources

ABS function - Microsoft
ABSOLUTE Function in Excel (ABS)
Absolute value - Wikipedia

'ABS' function examples

Functions in 'Math and trigonometry' category

The ABS function function is one of 62 functions in the 'Math and trigonometry' category.

Excel function categories

Excel categories

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