Understanding Percent Change and Percent Error

Percents are often used to describe how much a value has grown, shrunk, or how accurate a guess is. In these cases, we use percent change and percent error. Both concepts use a very similar formula: you compare a difference to the original (or actual) amount.

Percent Change (Increase and Decrease)

Percent change tells us the percentage by which a value has gone up (percent increase) or gone down (percent decrease) compared to its original amount.

The formula is:

${\text{Percent Change}} = \frac{\text{Amount of Change}}{\text{Original Amount}} \times 100\%$

To find the "Amount of Change," simply subtract the smaller number from the larger number to find the difference.

Example: Percent Increase

Problem: A price goes from \25$to$$30$. What is the percent increase?

1. Find the amount of change: $30 - 25 = 5$
2. Identify the original amount: The starting price was \25$.
3. Apply the formula: $\frac{5}{25} = 0.2$
4. Convert to a percentage: $0.2 \times 100\% = 20\%$

The price increased by $20\%$.

Percent Error

Percent error measures how inaccurate an estimate or measurement is compared to the exact, actual value. It tells you how far off you were as a percentage of the true value.

The formula is:

${\text{Percent Error}} = \frac{|\text{Estimated Value} - \text{Actual Value}|}{\text{Actual Value}} \times 100\%$

Note: The vertical bars $|$ mean absolute value, which just means the difference should always be a positive number.

Example: Percent Error

Problem: You estimated there were $48$ jellybeans in a jar, but the actual value is $50$. What is the percent error?

1. Find the difference (error): $|48 - 50| = 2$
2. Identify the actual value: The true number of jellybeans is $50$.
3. Apply the formula: $\frac{2}{50} = 0.04$
4. Convert to a percentage: $0.04 \times 100\% = 4\%$

Your estimate had a $4\%$ percent error.

Summary

Whether you are calculating percent change or percent error, the process is the same: find the difference, divide it by the original (or actual) number, and multiply by $100$ to turn it into a percentage!