Mathematics and Statistics in Detection

Three types of errors in nuclear medicine studies

User error, systematic error, and random (statistical) error.

User error

Mistakes made by the user.

Characteristics of user error

Occasional, can produce grossly inaccurate results, can be minimized but not eliminated.

Can user errors be completely eliminated?

No. They can be minimized, but they will always happen.

Systematic error

Predictable, preventable error that consistently produces incorrect or off measurements.

Potential cause of systematic error

Equipment malfunction.

Random error

Statistical error caused by variability from one measurement to another.

Can the user control random errors?

No.

Why random errors are always present in decay

Radioactive decay is fundamentally a random statistical process.

Type of process radioactive decay is

A random process.

What decay measurements are subject to

Statistical fluctuation.

Primary source of statistical fluctuation in nuclear medicine

Equipment used for measuring or imaging.

Predominant source of instrumentation errors

Statistical fluctuation.

Two applications of statistics in nuclear medicine

1. Verifying proper equipment operation.

2. Predicting accuracy of a single measurement.

How proper equipment operation is verified

By performing daily quality control (QC).

Importance of predicting single measurement accuracy

It helps verify that received information is accurate.

What variance and standard deviation describe in imaging

Errors in measurements or noise in images.

Noise in an image

Variation or statistical uncertainty in the measurements making up the image.

Two statistical concepts used to describe image noise

Variance and standard deviation.

Frequency distribution

A way to collect data and understand where numbers fall within it.

Example of frequency distribution in nuclear medicine

Number of bone scans performed per month in a department.

Mean

The average of all values.

How to calculate the mean

Sum all values and divide by the total number of values.

Median

The middle value in a set of measurements.

What the median indicates

Half of the measurements are larger and half are smaller.

Mode

The most frequently occurring value.

Variance (concept)

A measure of how spread out the data are around the mean.

Standard deviation (concept)

A measure of the spread of measurements around the mean.

Measure of dispersion

A value that quantifies the spread of a distribution.

Range

The spread from the smallest value to the largest value.

Variance (mathematical definition)

The average squared deviation from the mean.

Standard deviation (mathematical definition)

The square root of the variance.

What standard deviation indicates

The extent of deviation for a group of measurements as a whole.

Three measures of dispersion

Range, variance, and standard deviation.

Step 1 in calculating standard deviation

Calculate the mean of all values.

Step 2 in calculating standard deviation

Subtract the mean from each individual value.

Step 3 in calculating standard deviation

Square each deviation.

Step 4 in calculating standard deviation

Add up all of the squared deviations.

Step 5 in calculating standard deviation

Divide the sum of squared deviations by the number of values.

Value obtained after dividing squared deviations by N

Variance.

Final step in calculating standard deviation

Take the square root of the variance.

Value obtained after square-rooting variance

Standard deviation.

Basic sequence for standard deviation calculation

Mean → deviations → square deviations → sum → divide (variance) → square root (SD).

What standard deviation reveals about measurements

How far away from the mean measurements are, on average.

Indication of a small standard deviation

Measurements are close to the mean.

Indication of a large standard deviation

Measurements are widely spread around the mean.

Effect on SD when measurements cluster near mean

SD becomes small.

Effect on SD when measurements spread widely

SD becomes large.

Data percentage within 1 SD in normal distribution

68.3%.

Data percentage within 2 SDs in normal distribution

95%.

Data percentage within 3 SDs in normal distribution

99.7%.

Graphical representation of standard deviation

The width of the distribution peak.

Distribution shape as standard deviation increases

The distribution becomes wider.

Coefficient of Variation (CV)

The ratio of standard deviation to the mean.

What a higher CV indicates

A greater level of dispersion around the mean.

Unit expression for Coefficient of Variation

Percentage (%).

Formula for Coefficient of Variation (CV)

CV = (Standard Deviation / Mean) × 100

Quality of CV ≤ 5%

Good information.

Quality of CV between 10% and 20%

Good.

Quality of CV between 20% and 30%

Acceptable.

Quality of CV > 30%

Unacceptable.

Effect on dispersion as CV increases

Dispersion around the mean increases.

Three probability distributions in nuclear medicine

Binomial, Poisson, and Normal (Gaussian).

Binomial distribution

The simplest model where a trial has only two possible outcomes.

Two outcomes in a binomial trial

Success or failure.

Symbol p in binomial distribution

Probability of success.

Symbol q in binomial distribution

Probability of failure.

Equation for probability of failure (q)

q = 1 − p

Requirement for p in binomial trials

It must remain constant throughout the trials.

Requirement for successive binomial trials

They must be independent.

Binomial expected average successes equation

μ = np

What μ = np represents

Expected average number of successes.

Binomial predicted variance equation

σ² = npq

What σ² = npq represents

Predicted variance in binomial distribution.

Success in coin toss example

Heads.

Failure in coin toss example

Tails.

Values of p and q for fair coin toss

p = 0.5 and q = 0.5.

Why p and q equal 0.5 for a fair coin

Probability of success equals probability of failure.

Coin toss expected average successes (μ)

μ = 0.5n

Coin toss predicted variance (σ²)

σ² = 0.25n

Possible outcomes when rolling one die

6 outcomes.

Probability of rolling a 1 on a die (p)

p = 1/6 = 0.167

Probability of NOT rolling a 1 on a die (q)

q = 1 − p = 0.833

Expected average successes when rolling a die (μ)

μ = 0.167n

Predicted variance when rolling a die (σ²)

σ² = 0.139n

Two binomial outcomes for radioactive decay atom

Atom decays (p) or does not decay (q).

Fraction of radioactive material remaining after time t

e⁻λt

Value of p for radioactive decay

p = 1 − e⁻λt

Value of q for radioactive decay

q = e⁻λt

Poisson distribution

A special case of the binomial distribution with independent events.

Event independence in Poisson distribution

Each event must occur independently.

Number of possible events in a time period (Poisson)

An infinite number of events.

Relationship between p and time length (Poisson)

Probability of success is proportional to length of time period.

Behavior of p as time t → 0 in Poisson

p approaches 0.

Behavior of q as time t → 0 in Poisson

q approaches 1.

Valid value types for Poisson distribution

Non-negative integer values.

Poisson probability equation P(N;m)

P(N;m) = (e⁻ᵐ · mᴺ) / N!

Parameter that describes a Poisson distribution

The mean.

Relationship between variance and mean in Poisson

Variance equals the mean (σ² = μ).

Equation for Poisson variance

σ² = np (since q ≈ 1).