Study the following table of population in a district and the average population served per doctor. Calculate the harmonic mean:rn rnrnrnrnAREArnrnrnPOPULATION SERVED PER DOCTORrnrnrnPOPULATIONrnrnrnNUMBER OF DOCTORSrnrnrnrnrnRURALrnrnrn1000rnrnrn50000rnrnrn50rnrnrnrnrnURBANrnrnrn500rnrnrn50000rnrnrn100rnrnrnrnrnTOTALrnrn rnrn100000rnrnrn150rnrnrnrn

## **Core Concept**
The harmonic mean is a type of average, which is calculated by taking the reciprocal of the arithmetic mean of the reciprocals of the given numbers. It is often used for rates, such as population served per doctor, to avoid biases from uneven distributions.

## **Why the Correct Answer is Right**
To calculate the harmonic mean of the population served per doctor, we use the formula:
[ text{Harmonic Mean} = frac{sum N}{sum frac{N}{X}} ]
where ( N ) is the number of doctors and ( X ) is the population served per doctor.

For rural areas: ( N = 50 ), ( X = 1000 ), so ( frac{N}{X} = frac{50}{1000} = 0.05 )

For urban areas: ( N = 100 ), ( X = 500 ), so ( frac{N}{X} = frac{100}{500} = 0.2 )

[ sum N = 50 + 100 = 150 ]
[ sum frac{N}{X} = 0.05 + 0.2 = 0.25 ]

[ text{Harmonic Mean} = frac{150}{0.25} = 600 ]

## **Why Each Wrong Option is Incorrect**
- **Option A:** This option is incorrect because it doesn't match our calculated harmonic mean of 600.
- **Option B:** This option is incorrect for the same reason as Option A; the calculation does not support this value.
- **Option C:** Similarly, this option does not align with the correct calculation of the harmonic mean.
- **Option D:** This is the correct answer, so we don't discuss it here.

## **Clinical Pearl / High-Yield Fact**
In epidemiology and public health, the harmonic mean is particularly useful for averaging rates, such as the population served per doctor, because it gives more weight to smaller values, providing a more realistic estimate of the average service across different populations.

## **Correct Answer: B. 600**

✓ Correct Answer: C. 667