1. 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.

2. Definitions PopulationThe animals in a P.A., in a woreda or in a zone. TotalThe total number of animals in the population. MeanThe average number of cattle owned per household in the population. VarianceA measure of the variation in numbers of cattle owned by different households in the population. Standard errorThe precision with which the total number of animals is estimated. The standard error is calculated from the variance. Population Estimation

3. 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. 4.To calculate the standard error first calculate the variance s 2 from the sample of households. s 2 = Sum (y – m) 2 / (n –1) 5.The standard error is the square root of [ N(N - n)s 2 / 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 

5. 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 

6. Calculation of number of cattle together with standard error in Haro P.A. Herd sizelowmedi um high Cattle numbers 16 1012 15 37 1013 15 48 1013 15 58 1014 21 58 1015 22 58 10 15 5 Number (n) 7 12 11 Sample mean (m) 4.0 8.815.5 Standard deviation (s)1.53 1.423.17 No. households in village (N) 565 216 78 Nm226018901205Sum of Nm5355 N(N-n)s 2 /n10509074274785Sum of N(N-n)s 2 /n117303Square root342 So estimated number of cattle in village is 5355  342

7. 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?

8. Effect of selection of sample of households on precision of population estimate in Haro P.A. Size of householdEstimated total s.e.% reduction LowMediumHigh Households in P.A. (N) 56521678 Sample Actual (n) 7 12115355342 Proportional (n) 20 7 3535526223 Ideal (n) 16 7 7535525625

9. 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.

10. 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.

11. 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.