9.4 Percent Change & Basic Probability

Percent Change

Percent change tells you how much a value changed, relative to where it started. The formula is one line:

Percent change = (new − old) ÷ old × 100%

• New is the later or updated value.
• Old is the earlier or starting value.
• The denominator is always the old value — the one you started from.

A positive result is a percent increase. A negative result is a percent decrease.

Worked Example: Heart Rate Increase

A patient's heart rate goes from 80 bpm to 96 bpm after ambulation. What is the percent increase?

Step 1: Identify old and new. Old = 80, new = 96. Step 2: Subtract: 96 − 80 = 16. Step 3: Divide by old: 16 ÷ 80 = 0.20. Step 4: Convert to percent: 0.20 × 100 = 20%.

The heart rate increased by 20%.

Worked Example: Temperature Drop

A patient's temperature falls from 101.2 °F to 99.6 °F after an antipyretic. What is the percent decrease?

Step 1: Old = 101.2, new = 99.6. Step 2: Subtract: 99.6 − 101.2 = −1.6. Step 3: Divide by old: −1.6 ÷ 101.2 ≈ −0.0158. Step 4: Convert: −0.0158 × 100 ≈ −1.58% (about a 1.58% decrease).

The Base Trap — Why Percent Change Is Not Symmetric

A rise from 80 to 100:

• Old = 80, new = 100. Change = 20. Percent change = 20 ÷ 80 = 25% increase.

A fall from 100 to 80:

• Old = 100, new = 80. Change = −20. Percent change = −20 ÷ 100 = 20% decrease.

Same numbers, different base — so the percent is different. This is the single most common percent-change error: assuming that a rise and a fall between the same two values give the same percent. They do not.

Worked Example: Lab Value Shift

A patient's potassium goes from 4.0 mEq/L to 3.2 mEq/L. What is the percent change?

Step 1: Old = 4.0, new = 3.2. Step 2: Change = 3.2 − 4.0 = −0.8. Step 3: −0.8 ÷ 4.0 = −0.20. Step 4: −0.20 × 100 = −20% (a 20% decrease).

Basic Probability

Scope note: The NLN's published Data and Information outline names summary statistics (mean, median, mode), comparison of data sets, and interpretation of data plots and tables. It does not list probability. Treat this part as supporting background for reading proportions and "chance" wording in data items, not as a guaranteed content area — study it after you are solid on Sections 9.1 through 9.5.

Probability measures how likely an event is. For equally likely outcomes:

P(event) = favorable outcomes ÷ total outcomes

• The result is a number between 0 and 1 (or 0% and 100%).
• 0 means the event is impossible.
• 1 means the event is certain.

Worked Example: Drawing a Card

From a standard 52-card deck, what is the probability of drawing a heart?

Step 1: Count favorable outcomes — 13 hearts. Step 2: Count total outcomes — 52 cards. Step 3: Divide: 13 ÷ 52 = 0.25 = 25% (or "1 in 4").

Worked Example: Patient Outcomes

A small ward had 40 patients last week: 24 discharged home, 8 transferred to rehab, 6 transferred to ICU, 2 expired. What is the probability that a randomly chosen patient from this group was discharged home?

Step 1: Favorable (discharged home) = 24. Step 2: Total = 40. Step 3: 24 ÷ 40 = 0.60 = 60%.

The Complement Rule

If you know P(A), the probability that A does not happen is:

P(not A) = 1 − P(A)

In the ward example, the probability a patient was not discharged home is 1 − 0.60 = 0.40 (40%).

"And" and "Or" — Basic Intuition

At Grade 8 level, you only need the intuition, not the full formulas:

• "And" means both events must happen. The probability of two independent events both happening is smaller than either one alone — you multiply. (Example: the chance of two independent events each with probability 0.5 both happening is 0.5 × 0.5 = 0.25.)
• "Or" means at least one event happens. The probability is larger than either event alone.

The NEX will not ask you to combine complex compound events. If a question says "and," think multiplication and a smaller result. If it says "or," think addition (of non-overlapping events) and a larger result.

Common Traps

1. Wrong base in percent change. The denominator is the old value, not the new value. A change from 50 to 80 is (80 − 50) ÷ 50 = 60% increase, not (80 − 50) ÷ 80 = 37.5%.
2. Probability greater than 1. If your favorable count is larger than your total count, something is wrong — you cannot have more favorable outcomes than total outcomes. Recheck both counts.
3. Counting outcomes wrong. When counting total outcomes for a probability, make sure every outcome is counted once and only once. Double-counting a patient or missing a category shifts both numerator and denominator.
4. Mixing percent change with percentage points. A rate that goes from 10% to 14% is a 4 percentage-point increase, but the percent change is (14 − 10) ÷ 10 = 40% relative increase. These are different statements; do not confuse them.

Quick-Reference Summary