Act as a data analyst for mobile games. I will give you a question, and you will give me a succinct answer. If necessary, show your formula/homework for how you got your answer.

My first question: If I have a player retention curve of 53% on Day 1, 21% on Day 7, 17% on day 14, and 12% on Day 30, can you estimate how many days the average player will play?
user
Sure, I'd be happy to help you with that. The average lifespan of a player is typically estimated using retention rate, which is the percentage of players who return after a particular day. In your case, the average retention rate is given by:

Average Retention Rate = (1 - (1 - Retention Rate)^Days)

where Days is the number of days from the initial cohort. Here's how you can use this formula to answer your question:

1. Calculate the average Retention Rate:

   Average Retention Rate = (1 - (1 - 0.53)^1) + (1 - (1 - 0.21)^7) + (1 - (1 - 0.17)^14) + (1 - (1 - 0.12)^30)

         = 0.53 + 0.21 * (1 - 0.89) + 0.17 * 0.83 + 0.12 * 0.79

         = 0.88

2. Substitute the average Retention Rate into the formula:

   Average Lifespan = Days / Log(1 - Average Retention Rate / (1 - Average Retention Rate))

   Average Lifespan = 30 / Log(1 - 0.88 / (1 - 0.88))

       ≈ 20.18 Days

So, approximately, players stay for around 20.18 days. This estimate assumes that each player returns every day, which is unlikely; however, it's still a close representation of how long players remain.
assistant
Act as a data analyst for mobile games. I will give you a question, and you will give me a succinct answer. If necessary, show your formula/homework for how you got your answer.

My first question: If I have a player retention curve of 53% on Day 1, 21% on Day 7, 17% on day 14, and 12% on Day 30, can you estimate how many days the average player will play?
user
To estimate the average number of days a player will play, we can use the area under the retention curve. Since we only have discrete data points, we can approximate the area using the trapezoidal rule.

The trapezoidal rule involves summing the areas of trapezoids formed by consecutive data points, and then dividing by the interval between the points (in our case, the number of days between each retention measurement).

Here's the formula for the trapezoidal rule:
area = (1/2) \* (y1 + y2) \* (x2 - x1)

where (x1, y1) and (x2, y2) are consecutive data points.

We'll calculate the area for each pair of consecutive data points and then add up the areas to estimate the total area under the retention curve.

Area between Day 0 (100%) and Day 1 (53%):
area1 = (1/2) \* (100 + 53) \* (1 - 0)
area1 = 76.5

Area between Day 1 (53%) and Day 7 (21%):
area2 = (1/2) \* (53 + 21) \* (7 - 1)
area2 = 168

Area between Day 7 (21%) and Day 14 (17%):
area3 = (1/2) \* (21 + 17) \* (14 - 7)
area3 = 39

Area between Day 14 (17%) and Day 30 (12%):
area4 = (1/2) \* (17 + 12) \* (30 - 14)
area4 = 102.5

Now we add up all the areas:
total\_area = area1 + area2 + area3 + area4
total\_area = 76.5 + 168 + 39 + 102.5
total\_area = 386

Since the total area under the retention curve represents the average number of days a player will play, our estimation is approximately **386 days**.

However, this estimation assumes a linear decay between the given retention data points. In reality, the retention curve might not be perfectly linear between the given points, and we'd need more data to get a more accurate estimate.

Additionally, this method gives a relatively high estimate for the average number of days a player will play because it takes into account the long tail of low retention percentages. In practice, you might want to focus on a specific time frame (e.g., the first 90 days) to get a more practical understanding of player engagement.
assistant