Statistics: mean, median, mode, SD, CV and confidence intervals

9 ASCP MLS practice questions on Statistics: mean, median, mode, SD, CV and confidence intervals, part of Laboratory Operations. The first 9 are below; the first 5 with answers and explanations (tap Show answer after you have chosen), the rest in Learn with a free account.

1. Question 1 · Hard

   A laboratory establishes its own adult reference interval for serum GGT from 120 healthy volunteers, after excluding people with liver disease, heavy alcohol use or enzyme-inducing drugs. Summary of the results: mean 31 U/L, SD 18 U/L, median 25 U/L, mode 21 U/L; lowest value 6 U/L, highest 112 U/L; 2.5th percentile 9 U/L, 97.5th percentile 78 U/L. A histogram shows a long tail toward high values. A colleague proposes reporting the mean ± 2 SD. Which reference interval should the laboratory adopt?

   1. A6 to 112 U/L, the full range of the 120 results
   2. B9 to 78 U/L, from the 2.5th and 97.5th percentiles
   3. C0 to 67 U/L, the mean ± 2 SD with negatives set to 0
   4. D13 to 49 U/L, the mean ± 1 SD of the results

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   Answer: B · 9 to 78 U/L, from the 2.5th and 97.5th percentiles

   The mean is higher than the median, which is higher than the mode, and the mean minus 2 SD (31 - 36 = -5 U/L) falls below zero, so the data are skewed to the right and not Gaussian. Parametric limits would give an impossible lower limit and an upper limit of 67 U/L that labels too many healthy people as high. CLSI EP28 recommends the nonparametric method for such data: the 2.5th and 97.5th percentiles of at least 120 reference subjects form the central 95% interval. Alternatively, the data could be log-transformed to a near-Gaussian shape before parametric limits are calculated.

   • A: The full range includes the extreme values and covers more than the central 95% of healthy people.
   • C: The mean ± 2 SD assumes Gaussian data; here it gives a negative lower limit and cuts off healthy high values.
   • D: The mean ± 1 SD covers only about two-thirds of healthy people, even in Gaussian data.

2. Question 2 · Medium

   The histogram shows the turnaround times of 600 STAT troponin tests. Lines 1 to 4 mark the mode, the median, the mean and the 90th percentile, in some order. The mean is marked by:

   

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   1. ALine 1
   2. BLine 2
   3. CLine 3
   4. DLine 4

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   Answer: C · Line 3

   The distribution is skewed to the right: most tests finish in 20 to 40 minutes, but a tail of delayed samples runs past 100 minutes. The mode sits at the peak (line 1, about 28 minutes) and the median splits the tests in half (line 2, 38 minutes), while the tail pulls the mean to the right of the median (line 3, about 44 minutes). Line 4, far out in the tail at about 77 minutes, is the 90th percentile. Turnaround time is usually monitored by the median and the 90th percentile, because a few very late samples inflate the mean.

   • A: Line 1 is at the peak of the histogram: the mode.
   • B: Line 2 is the median, which the tail pulls less than the mean.
   • D: Line 4 is far out in the tail: the 90th percentile.

3. Question 3 · Medium

   Sodium results in a healthy reference group follow a Gaussian distribution with a mean of 140 mmol/L and an SD of 2 mmol/L. Which range contains about 95% of the results?

   1. A136 to 144 mmol/L
   2. B134 to 146 mmol/L
   3. C138 to 142 mmol/L
   4. D137 to 143 mmol/L

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   Answer: A · 136 to 144 mmol/L

   In a Gaussian distribution about 68% of values lie within ±1 SD, 95% within ±2 SD and 99.7% within ±3 SD. Here ±2 SD is 140 ± 4, or 136 to 144 mmol/L. This central 95% is how a parametric reference interval is set, so about 5% of healthy people fall outside it. Strictly, 95% corresponds to ±1.96 SD.

   • B: 134 to 146 mmol/L is ±3 SD, which holds about 99.7% of results.
   • C: 138 to 142 mmol/L is ±1 SD, which holds about 68% of results.
   • D: 137 to 143 mmol/L is ±1.5 SD, which holds about 87% of results.

4. Question 4 · Hard

   Five replicate total CO2 results are 24, 26, 28, 27 and 25 mmol/L, with a mean of 26 mmol/L. The sample standard deviation is:

   1. A1.41 mmol/L
   2. B1.20 mmol/L
   3. C2.50 mmol/L
   4. D1.58 mmol/L

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   Answer: D · 1.58 mmol/L

   The deviations from the mean are -2, 0, +2, +1 and -1, and their squares add up to 4 + 0 + 4 + 1 + 1 = 10. For a sample, this sum is divided by n - 1 = 4, giving a variance of 2.5. The SD is the square root of the variance: √2.5 = 1.58 mmol/L. Dividing by n - 1 rather than n corrects for using the sample's own mean.

   • A: 1.41 mmol/L divides by n (5) instead of n - 1.
   • B: 1.20 mmol/L is the mean absolute deviation, not the SD.
   • C: 2.50 is the variance; its square root has not been taken.

5. Question 5 · Medium

   A TSH method's low control has a mean of 0.50 mIU/L and an SD of 0.04 mIU/L; its high control has a mean of 20.0 mIU/L and an SD of 1.2 mIU/L. Which level shows better relative precision?

   1. AThe high level, because its CV is lower
   2. BNeither, because the two means differ too much
   3. CThe low level, because its SD is smaller
   4. DThe low level, as higher levels are less precise

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   Answer: A · The high level, because its CV is lower

   CV = SD ÷ mean × 100, which expresses imprecision relative to the concentration. The low control's CV is 0.04 ÷ 0.50 × 100 = 8.0%, and the high control's is 1.2 ÷ 20.0 × 100 = 6.0%. The high level is therefore relatively more precise, although its SD is 30 times larger. SD usually rises with concentration, which is why CV is used to compare precision across levels.

   • B: Comparing levels with different means is exactly what CV is for.
   • C: A smaller SD at a much lower mean can still mean poorer relative precision.
   • D: Relative precision often improves at higher levels, as it does here.

6. Question 6 · Hard

   Twenty-five replicate chloride measurements give a mean of 100 mmol/L and an SD of 2.5 mmol/L. What is the approximate 95% confidence interval for the mean?

   1. A99.8 to 100.2 mmol/L
   2. B95.0 to 105.0 mmol/L
   3. C99.0 to 101.0 mmol/L
   4. D97.5 to 102.5 mmol/L

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7. Question 7 · Medium

   Potassium results on six outpatient samples are 3.6, 4.0, 4.2, 4.6, 4.8 and 8.4 mmol/L. The median is:

   1. A4.9 mmol/L
   2. B6.0 mmol/L
   3. C4.4 mmol/L
   4. D4.2 mmol/L

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8. Question 8 · Hard

   How is a reference interval usually established for a new test?

   1. AThe highest and lowest values in hospital patients
   2. BMean ± 1 SD of results from 20 hospital patients
   3. CThe manufacturer's interval, adopted without checking
   4. DThe central 95% of results from at least 120 healthy people

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9. Question 9 · Easy

   How is the coefficient of variation (CV) calculated?

   1. AMean ÷ standard deviation
   2. BStandard deviation squared
   3. CStandard deviation ÷ mean × 100
   4. D(Highest value - lowest value) ÷ mean

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