Population Estimation

Objective : To estimate from a sample of households the numbers of

animals in a population and to provide a measure of

precision for the estimate.

Assumption : Units in the population are selected at random.

Population : For our purposes we can consider this to be the animals

in a P.A. or the animals in a woreda. Estimation of

population numbers at the zone level is more

problematic since the selection of woredas was

representative, not random.

Definitions

Population The animals in a P.A., in a woreda or in a zone.

Total The total number of animals in the population.

Mean The average number of cattle owned per household in

the population.

Variance A measure of the variation in numbers of cattle owned by different households in the population.

Standard error The precision with which the total number of animals is estimated. The standard error is calculated from the variance.

Population Estimation

A simple example of the estimation of population numbers

Random sample of household in a P.A.

1. Suppose that n households are sampled from N households in a P.A.

2. To get sample mean add together the numbers of animals in the sampled households and divide by n. Write the sample mean as m.

3. Multiply m by N to get estimate of the total number of animals in the P.A. = Nm.

4. To calculate the standard error first calculate the variance s2 from the sample of households.

s2 = Sum (y – m)2 / (n –1)

5. The standard error is the square root of

[ N(N - n)s2 / n ]

6. The estimated number of cattle in the P.A. is then

A simple example of the estimation of population numbers (continued)

/nn)sN(NNm 2

Stratified and clustered sampling

• Methods get more complicated but the principle is the same. Aim is to estimate a population total at the P.A. or woreda level and to use the variations observed among households or P.A.s to obtain standard error for the total.

• For example, the estimated number of cattle in a P.A. stratified by household size is

where summation is over strata.

• Note that the population numbers of households N for each of the strata with low, medium and high numbers of livestock are needed in the above formulae.

]/nn)sN(N[SumNmSum 2

Calculation of number of cattle together with standard error in Haro P.A.

Herd size low medium

high

Cattle numbers 1 6 10 12 15

3 7 10 13 15

4 8 10 13 15

5 8 10 14 21

5 8 10 15 22

5 8 10 15

5

Number (n) 7 12 11

Sample mean (m) 4.0 8.8 15.5

Standard deviation (s) 1.53 1.42 3.17

No. households in village (N) 565 216 78

Nm 2260 1890 1205 Sum of Nm 5355

N(N-n)s2/n 105090 7427 4785 Sum of N(N-n)s2/n 117303 Square root 342

So estimated number of cattle in village is 5355 342

So what can we say about the number of cattle in Haro P.A.?

• The estimated number is 5355 cattle.

• The s.e. (which measures the precision with which the number is estimated) is 342 cattle.

• If we multiply the s.e. by 2 we can say that the total number lies in the range

5355 – 2 x 342 to 5355 + 2 x 342 or 4671 to 6039 cattle with a 95% chance of being correct.

Question 1. Is this range reasonable or unreasonable?

Question 2. How can we reduce the range?

Effect of selection of sample of households on precision of population estimate in Haro P.A.

Size of household Estimated total

s.e. % reduction

Low Medium High

Households in P.A. (N) 565 216 78

Sample

Actual (n) 7 12 11 5355 342

Proportional (n) 20 7 3 5355 262 23

Ideal (n) 16 7 7 5355 256 25

Calculation of estimate of number of cattle in a woreda

• Calculation is similar to that at P.A. level.

• Instead of numbers of cattle per household we use estimated numbers of cattle per P.A.

• Instead of numbers of households we use numbers of P.A.s in the woreda.

• The s.e. is now based on variation both among P.A.s within the woreda and among households within the P.A.

• Again we need to consider ways of minimising this s.e.

Improving the precision of population estimation at the woreda level

• As for households within a P.A. one can consider stratification of P.A.s in the woreda into groups of P.A.s likely to have similar livestock densities.

• One can also consider stratification of the woreda by agro-ecological zone.

• Determine, or have available, the number of households for P.A.s both sampled and not sampled in the woreda.

Conclusions

• Decide how important it is to calculate estimates of numbers of cattle in a population.

• If it is important, then pay very careful attention to the sampling design.

• Use knowledge gained from previous surveys to determine likely levels of variation in livestock numbers from household to household within a P.A. and from P.A. to P.A. in a woreda.

• Consider the types of stratification that might be applied to reduce these variations.

• Use the population estimation formulae to compare the effect of different sample sizes on the likely precision of a population total.