Counting Statistics and the Poisson distribution

In the process of flow cytometry cells (events) are presented for analysis at random and their distribution will be described by Poisson statistics. Any sub-set of cells will also be distributed at random within the parent population and we can consider the statistics of these separately. The essential feature of Poisson distributions is that if N events are observed the standard deviation (SD) associated with that count is square root of N . The coefficient of variation (CV) is then given by

CV = 100 X SD/ N or 100 / sqrt N.

This has two important consequences for determining how much data to collect. If a sub-set smaller by a factor of S is investigated the total number of events processed must be increased by a factor S to maintain the same precision. To improve precision by a factor P the number of rare events recorded, and hence the total number of analysed, must be increased by a factor of P2.

Consider a data set collected for 10000 cells, if a sub-population exists at the 10% level 1,000 of these would be observed with an associated CV of 3.16%, which would be acceptable for many experiments. However, at the 1% level only 100 cells would be expected with a CV of 10% and a drop in precision. This can only be improved by counting more cells and it is useful to construct a reference chart for estimating how many events to process for a required precision.

For a CV of (%) |
1.0 | 2.5 | 5.0 | 10 | 20 | |

Positives required |
10,000 | 1,600 | 400 | 100 | 25 | |

At a frequency of |
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% |
1 in |
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10 | 10 | 100,000 | 16,000 | 4,000 | 1,000 | 250 |

1 | 100 | 1000,000 | 160,000 | 40,000 | 10,000 | 2,500 |

.1 | 1,000 | 10,000,000 | 1,600,000 | 400,000 | 100,000 | 25,000 |

.01 | 10,000 | 10,0,000,000 | 16,000,000 | 4,000,000 | 1,000,000 | 250,000 |

.001 | 100,000 | 1000,000,000 | 160,000,000 | 40,000,000 | 10,000,000 | 2,500,000 |

The table presents such a chart, a total count of 10 7 events which represents about 20 minutes per sample at 10,000 events per second. It is limited to a frequency of 1 in 100,000 beyond which only rough estimates can be obtained even after collecting 10 7 events. Naturally, the sub-set has to be well resolved if these theoretical levels are to be approached in which case the rule is: the number of rare cells examined determines accuracy.

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