Part 1: Introduce
🔹 Definition: The POISSON.DIST function in Microsoft Excel calculates the Poisson distribution, which measures how often an event is likely to occur within a specified period.
🔹 Purpose: It’s used to predict the probability of certain events when you know the average number of times the event has occurred. It’s beneficial for rare events in large datasets.
🔹 Syntax & Arguments:
POISSON.DIST(x, mean, cumulative) 🔹 Explain the Arguments in the function:
x: The actual number of events.mean: The average number of times the event occurs over a specified period.cumulative: A logical value; TRUE returns the cumulative distribution function; FALSE returns the probability mass function.
🔹 Return value: This function returns the Poisson distribution probability.
🔹 Remarks: The POISSON.DIST function can be used for rare events, given a large sample size. Ensure that the mean is a positive number.
Part 2: Examples
🌟 Example 1:
• Purpose of example: Determine the probability of receiving exactly 3 customer complaints daily when the average number of complaints is 2.
• Data tables and formulas:
| A | B | C | D | |
|---|---|---|---|---|
| 1 | x | Mean | Cumulative | Result |
| 2 | 3 | 2 | FALSE | =POISSON.DIST(A2,B2,C2) |
| 3 | 0.180 |
• Explanation: Given an average of 2 complaints per day, the probability of receiving exactly 3 complaints on a particular day is 18.0%.
🌟 Example 2:
• Purpose of example: Determine the cumulative probability of receiving up to 4 faulty products in a shipment when the average number of defective products is 3.
• Data tables and formulas:
| A | B | C | D | |
|---|---|---|---|---|
| 1 | x | Mean | Cumulative | Result |
| 2 | 4 | 3 | TRUE | =POISSON.DIST(A2,B2,C2) |
| 3 | 0.647 |
• Explanation: Given an average of 3 faulty products per shipment, the cumulative probability of receiving up to 4 faulty products is 64.7%.
🌟 Example 3:
• Purpose of example: Determine the probability of a website receiving precisely 10 hits in an hour when the average number of hits per hour is 8.
• Data tables and formulas:
| A | B | C | D | |
|---|---|---|---|---|
| 1 | x | Mean | Cumulative | Result |
| 2 | 10 | 8 | FALSE | =POISSON.DIST(A2,B2,C2) |
| 3 | 0.112 |
• Explanation: Given an average of 8 website hits per hour, the probability of receiving exactly 10 hits per hour is 11.2%.
🌟 Example 4:
• Purpose of example: Determine the cumulative probability of a call center receiving up to 5 calls in 10 minutes when the average number of calls per 10 minutes is 4.
• Data tables and formulas:
| A | B | C | D | |
|---|---|---|---|---|
| 1 | x | Mean | Cumulative | Result |
| 2 | 5 | 4 | TRUE | =POISSON.DIST(A2,B2,C2) |
| 3 | 0.785 |
• Explanation: Given an average of 4 calls every 10 minutes, the cumulative probability of receiving up to 5 calls in that time frame is 78.5%.
🌟 Example 5:
• Purpose of example: Determine the probability of a factory producing exactly 7 defective items daily when the average number of faulty items made daily is 5.
• Data tables and formulas:
| A | B | C | D | |
|---|---|---|---|---|
| 1 | x | Mean | Cumulative | Result |
| 2 | 7 | 5 | FALSE | =POISSON.DIST(A2,B2,C2) |
| 3 | 0.104 |
• Explanation: Given an average production of 5 defective items daily, the probability of the factory producing exactly 7 defective items on a particular day is 10.4%.
🌟 Example 6: Using POISSON.DIST with IF
• Purpose of example: Determine if a server faces 15 requests per minute on average and is under unusual load if it receives 25 recommendations in a minute.
• Data tables and formulas:
| A | B | C | D | E | |
|---|---|---|---|---|---|
| 1 | x | Mean | Cumulative | Formula | Result |
| 2 | 25 | 15 | TRUE | =IF(POISSON.DIST(A2,B2,C2)>0.95, “Unusual”, “Normal”) | Unusual |
• Explanation: If the cumulative probability of receiving 25 or fewer requests is greater than 95%, it’s considered unusual. In this case, the server receiving 25 recommendations in a minute is deemed unique.
🌟 Example 7: Using POISSON.DIST with SUM
• Purpose of example: Calculate the combined probability of a bookstore selling 2, 3, or 4 rare books daily when the average sale is 3 rare books daily.
• Data tables and formulas:
| A | B | C | D | E | |
|---|---|---|---|---|---|
| 1 | x | Mean | Cumulative | Formula | Result |
| 2 | 2 | 3 | FALSE | =SUM(POISSON.DIST(A2,B2,C2), POISSON.DIST(A3,B3,C3), POISSON.DIST(A4,B4,C4)) | 0.647 |
| 3 | 3 | 3 | FALSE | ||
| 4 | 4 | 3 | FALSE |
• Explanation: The individual probabilities of selling 2, 3, or 4 rare books are summed up to get a combined probability. The combined chance of selling 2 to 4 rare books daily is 64.7%.
🌟 Example 8: Using POISSON.DIST with VLOOKUP
• Purpose of example: Given a table of average sales, determine the probability of selling 5 products of a specific type per day.
• Data tables and formulas:
| A | B | C | D | E | |
|---|---|---|---|---|---|
| 1 | Product Type | x | Cumulative | Formula | Result |
| 2 | Type A | 5 | FALSE | =POISSON.DIST(B2,VLOOKUP(A2,F:G,2,FALSE),C2) | 0.175 |
| 3 |
Average Sales Table:
| F | G | |
|---|---|---|
| 1 | Product Type | Average Sales |
| 2 | Type A | 4 |
| 3 | Type B | 6 |
• Explanation: The VLOOKUP function fetches the average “Type A” sales from the Average Sales Table. The POISSON.DIST function then calculates the probability of selling exactly 5 “Type A” products in a day. The result is a 17.5% chance.
🌟 Example 9: Using POISSON.DIST with AVERAGE
• Purpose of example: Calculate the average probability of a bakery selling 3, 4, or 5 pastries in an hour when the average sale is 4 pastries per hour.
• Data tables and formulas:
| A | B | C | D | E | |
|---|---|---|---|---|---|
| 1 | x | Mean | Cumulative | Formula | Result |
| 2 | 3 | 4 | FALSE | =AVERAGE(POISSON.DIST(A2,B2,C2), POISSON.DIST(A3,B3,C3), POISSON.DIST(A4,B4,C4)) | 0.238 |
| 3 | 4 | 4 | FALSE | ||
| 4 | 5 | 4 | FALSE |
• Explanation: The individual probabilities of selling 3, 4, or 5 pastries are averaged. The average probability of selling between 3 to 5 pastries in an hour, given an average sale of 4 pastries, is 23.8%.
🌟 Example 10: Using POISSON.DIST with MAX
• Purpose of example: Determine the highest probability of a website receiving 10, 11, or 12 visitors in an hour when the average number of visitors is 10.
• Data tables and formulas:
| A | B | C | D | E | |
|---|---|---|---|---|---|
| 1 | x | Mean | Cumulative | Formula | Result |
| 2 | 10 | 10 | FALSE | =MAX(POISSON.DIST(A2,B2,C2), POISSON.DIST(A3,B3,C3), POISSON.DIST(A4,B4,C4)) | 0.125 |
| 3 | 11 | 10 | FALSE | ||
| 4 | 12 | 10 | FALSE |
• Explanation: The MAX function is used to determine the highest probability among the three scenarios. The highest probability is for the website to receive precisely 10 visitors in an hour, which is 12.5%.
🌟 Example 11: Using POISSON.DIST with MIN
• Purpose of example: Determine the lowest probability of a call center receiving 6, 7, or 8 calls in 10 minutes when the average number is 7.
• Data tables and formulas:
| A | B | C | D | E | |
|---|---|---|---|---|---|
| 1 | x | Mean | Cumulative | Formula | Result |
| 2 | 6 | 7 | FALSE | =MIN(POISSON.DIST(A2,B2,C2), POISSON.DIST(A3,B3,C3), POISSON.DIST(A4,B4,C4)) | 0.137 |
| 3 | 7 | 7 | FALSE | ||
| 4 | 8 | 7 | FALSE |
• Explanation: The MIN function is used to determine the lowest probability among the three scenarios. The lowest probability is for the call center to receive 6 calls in 10 minutes, which is 13.7%.
🌟 Example 12: Using POISSON.DIST with ROUND
• Purpose of example: Calculate the rounded probability of a factory producing 5 defective items daily when the average number of faulty items is 4.
• Data tables and formulas:
| A | B | C | D | E | |
|---|---|---|---|---|---|
| 1 | x | Mean | Cumulative | Formula | Result |
| 2 | 5 | 4 | FALSE | =ROUND(POISSON.DIST(A2,B2,C2), 3) | 0.157 |
• Explanation: The ROUND function is used to round the result of the POISSON.DIST function to three decimal places. The rounded probability of the factory producing exactly 5 defective items in a day, given an average of 4 faulty items, is 15.7%.
Part 3: Tips and tricks:
- 📌 The
POISSON.DISTfunction is handy for rare events in large datasets. - 📌 Ensure that the mean value you provide is non-negative. A negative mean will result in an error.
- 📌 Remember that the Poisson distribution assumes each event is independent of the others.
- 📌 For a visual representation, consider plotting the Poisson distribution using Excel’s charting tools.
- 📌 When using the
POISSON.DISTfunction, ensure your dataset is large enough to provide a meaningful result. - 📌 The
POISSON.DISTfunction can be used to model random and independent events, such as the number of phone calls to a call center or the number of emails received in an hour. - 📌 If you’re interested in the number of events over a different time frame or space, adjust the mean value accordingly.
- 📌 For more advanced statistical analysis, consider using other distribution functions in Excel or specialized statistical software.