A nephrology research group is establishing institutional reference values for serum ioniz… — USMLE Step 3

A nephrology research group is establishing institutional reference values for serum ionized calcium using a new ion-selective electrode method. They analyze fasting samples from $2,500$ healthy volunteers. The measurements are normally distributed with a mean of $4.80 \text{ mg/dL}$ and a variance of $0.04 \text{ (mg/dL)}^2$ . The laboratory director wants to define the central reference interval that encompasses approximately $95\%$ of the population. Based on the empirical rule, which of the following ranges represents this interval?

A
$4.20 \text{ to } 5.40 \text{ mg/dL}$
$4.40 \text{ to } 5.20 \text{ mg/dL}$ Answer
C
$4.60 \text{ to } 5.00 \text{ mg/dL}$
D
$4.72 \text{ to } 4.88 \text{ mg/dL}$
E
$4.76 \text{ to } 4.84 \text{ mg/dL}$

Answer

The expected range for

95\%

of the population is

4.40 \text{ to } 5.20 \text{ mg/dL}

, which represents the mean plus or minus two standard deviations.

The empirical rule (68-95-99.7 rule) states that for a normally distributed variable, approximately

95\%

of the data falls within two standard deviations of the mean. The vignette provides the variance as

0.04

. Because standard deviation is the square root of the variance, the standard deviation is

0.20 \text{ mg/dL}

. To find the

95\%

interval, calculate the mean plus or minus two standard deviations:

4.80 \pm 2(0.20) = 4.80 \pm 0.40

. This yields an interval of

4.40 \text{ to } 5.20 \text{ mg/dL}

Step-by-Step Solution

Identify the given statistical parameters from the vignette.

Mean =

4.80 \text{ mg/dL}

, Variance =

0.04 \text{ (mg/dL)}^2

These are the core descriptive statistics required to calculate population intervals.

Convert the variance to standard deviation.

Standard deviation =

\sqrt{0.04} = 0.20 \text{ mg/dL}

The empirical rule applies standard deviations, not variance, to determine data spread.

Apply the empirical rule to find the

95\%

interval.

The

95\%

interval is mean

\pm

2

standard deviations.

In a normal distribution, approximately

68\%

falls within

1

SD,

95\%

within

2

SD, and

99.7\%

within

3

SD.

Calculate the final boundaries.

Lower bound:

4.80 - 2(0.20) = 4.40

. Upper bound:

4.80 + 2(0.20) = 5.20

This establishes the precise interval for the requested

95\%

of the population.

Key Concept

The empirical rule (68-95-99.7 rule) dictates that in a normal distribution, approximately 95% of the data falls within two standard deviations of the mean. Variance must always be converted to standard deviation (by taking the square root) before applying this rule.

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