# Technologies for Forecasting Tree Fruit Load and Harvest Timing--From Ground, Sky and Time
## 2 Measurement of Fruit Number
### Manual Fruit Count  
#### 2.3.1 Sample Size and Variance  
The manual count of fruit for estimation of orchard crop load is a labor intensive and tedious task. Counting is typically restricted to a sample of units (trees or sections of rows) to reduce effort, but the number and selection of these sample units must be representative of the orchard in order to achieve an accurate assessment of orchard fruit load. [PERSON] [13] provides details on sampling strategies and statistics. Under random sampling (without replacement), the sample mean \(\overline{x}\) is an unbiased estimator of the population mean \(\mu\). The variance of the estimator is given by  
\[var(\overline{x})=\frac{(N-n)\sigma^{2}}{Nn}=\frac{(1-f)\sigma^{2}}{n} \tag{1}\]  
where \(n\) = sample size, \(N\) = population size, \(f\) ='sampling fraction' (the inverse probability) and \(\sigma\) = population standard deviation. An unbiased estimator of this variance is obtained by replacing the population standard deviation by the sample standard deviation, \(SD\)[13]. It may be intuitively recognized that the greater the variation between units, the more units that should be included in the estimate. The required sample size depends on population variability and not population size, although in practice, a larger population also involves a larger area, and thus potential for higher variability in growing conditions and plant history, and thus in yield. If the population size \(N\) is large compared to the sample size \(n\), the 'power calculation' can be used to estimate the required sample size \(n\) to achieve an estimate of the mean or the total of a population from a random sample within an acceptable error (\(e\)) by  
\[n=\left(\frac{z}{e}\right)^{2} \tag{2}\]  
where \(z\) denotes the z score for the upper \(\alpha/2\) point of the normal distribution, \(SD\) is sample standard deviation and \(e\) is the maximal allowable error (i.e., difference between  
Figure 2: Images of canopies associated with mango fruit loads of (**a**) 60, (**b**) 40 and (**c**) 8 tonnes per ha, with tree density of 400 trees/ha. the estimate and the true value). This relationship can be restated in terms of the coefficient of variation (\(CV\)) and maximal percentage error (PE), as  
\[n=\left(\frac{z\,\,CV}{e}\right)^{2} \tag{3}\]  
where \(CV=\)100 \(SD/\overline{x}\) (%), PE = 100 \(e/\overline{x}\) (%) and \(\overline{x}\) is the mean. As the sample size, \(n\), approaches the population size, \(N\), the equation for the sample size includes the finite population correction (FPC) = \((N-n)/\,\,N=1-f\), and the required sample size can be written as  
\[n^{*}=\frac{n\,\,N}{N+n} \tag{4}\]  
where \(n\) is calculated using Equation (2) or Equation (3). Equation (4) is typically employed if \(n\) is 5% or more of \(N\) (Table 3). These calculations require an estimate of population \(\sigma\), which may be available from prior experience or may involve a 'double sampling' strategy. In this strategy, a preliminary, smaller sample of size \(n_{1}\) is taken and the sample \(SD\) is used to estimate (approximately) the necessary sample size \(n\) using Equations (2)-(4). Then, a further sample of size \((n-n_{1})\) is taken from the remaining unsampled \((N-n_{1})\) trees. A simple jackknife-like error estimation can also be employed by splitting the count data into two sets and computing the variance of the two estimates to roughly predict the variance and its rate of decrease with \(n\) (see [14; 15]). In practical use, a 90% confidence limit, i.e., \(z=1.64\), is reasonable to apply in Equations (2) and (3). With this limit, a population with a coefficient of error (CE) = standard error of the mean (SEM)/mean of ~6% will have 90% of realizations fall within SEM \(\times\) z = 6.1\(\times\)1.64 = 10% of the mean value, and 30% of realizations within 5%. By way of examples, the estimated \(SD\) of fruit numbers per tree for a subsample of 18 trees from each orchard varied between 34 and 160 across 10 mango orchards (Table 3).