Stratified sampling

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Stratified sampling strategies[edit]


Stratified sampling ensures that at least one observation is picked from each of the strata, even if probability of it being selected is close to 0. Hence the statistical properties of the population may not be preserved if there are thin strata. A rule of thumb that is used to ensure this is that the population should consist of no more than six strata, but depending on special cases the rule can change - for example if there are 100 strata each with 1 million observations, it is perfectly fine to do a 10% stratified sampling on them.[citation needed]



The reasons to use stratified sampling rather than simple random sampling include[1]





Stratified sampling is not useful when the population cannot be exhaustively partitioned into disjoint subgroups. It would be a misapplication of the technique to make subgroups' sample sizes proportional to the amount of data available from the subgroups, rather than scaling sample sizes to subgroup sizes (or to their variances, if known to vary significantly e.g. by means of an F Test). Data (at the same time) to the subgroups' sizes within the total population. For an efficient way to partition sampling resources among groups that vary in their means, their variances, and their costs, see "optimum allocation". The problem of stratified sampling in the case of unknown class priors (ratio of subpopulations in the entire population) can have deleterious effect on the performance of any analysis on the dataset, e.g. classification.[2] In that regard, minimax sampling ratio can be used to make the dataset robust with respect to uncertainty in the underlying data generating process.[2]

Mean and Variance[edit]

The mean and variance of stratified random sampling is given by,[1]

\mu_s = \frac{1}{N} \sum_{h=1}^L N_h \mu_h

\sigma_s = \sum_{h=1}^L \left(\frac{N_h}{N}\right)^2 \left(\frac{N_h - n_h}{N_h}\right)\frac{\sigma_h^2}{n_h}


N = Size of entire population, should equal to sum of all strata size

N_h = Size of each strata

n_h = Number of observations in each strata

L = Count of strata

\sigma_h = sample standard deviation of strata h

\mu_h = sample mean of strata h

Strata Size Calculation[edit]

In general the size of the sample in each stratum is taken in proportion to the size of the stratum. This is called proportional allocation. Suppose that in a company there are the following staff:[3]

  • male, full-time: 90
  • male, part-time: 18
  • female, full-time: 9
  • female, part-time: 63
  • Total: 180


The first step is to find the total number of staff (180) and calculate the percentage in each group.

  • % male, full-time = 90 ÷ 180 = 50%
  • % male, part-time = 18 ÷ 180 = 10%
  • % female, full-time = 9 ÷ 180 = 5%
  • % female, part-time = 63 ÷ 180 = 35%

This tells us that of our sample of 40,

  • 50% should be male, full-time.
  • 10% should be male, part-time.
  • 5% should be female, full-time.
  • 35% should be female, part-time.
  • 50% of 40 is 20.
  • 10% of 40 is 4.
  • 5% of 40 is 2.
  • 35% of 40 is 14.


  • male, full-time = 90 × (40 ÷ 180) = 20
  • male, part-time = 18 × (40 ÷ 180) = 4
  • female, full-time = 9 × (40 ÷ 180) = 2
  • female, part-time = 63 × (40 ÷ 180) = 14

See also[edit]


  1. ^ a b "6.1 How to Use Stratified Sampling | STAT 506". Retrieved 2015-07-23. 
  3. ^ Hunt, Neville; Tyrrell, Sidney (2001). "Stratified Sampling". Webpage at Coventry University. Retrieved 12 July 2012. 

Further reading[edit]

  • Särndal, Carl-Erik; et al. (2003). "Stratified Sampling". Model Assisted Survey Sampling. New York: Springer. pp. 100–109. ISBN 0-387-40620-4.