We're probably all a little exhausted with blog posts regarding Apple (NASDAQ:AAPL). We have been beaten to a pulp with figures and conjecture that provide little in the way of objective data to support the claims. Well, here's another one, except I will show the data, method and calculations. First, let's start with the synopsis, so that if you don't feel like digging through this, you can bail soon. Apple will probably report for Q1 2013 around 69.1M iPhones/iPads sold with revenue of $58B and profit of $14B. The upside surprise could reach 83M iPhones/iPads, with revenue of $70B and profit of $17B. Looking toward March, the share price should reach and average a closing price of at least $650. These figures should sound all too familiar by now. What follows is the method to the madness:

For those not well versed in statistics, we will begin with a super quick primer. If you have seen all this before then skip it. In doing analysis, the first calculation after deriving an average is the standard deviation. The standard deviation is a measure of the variability within a data set. Given a normal distribution of data, 95% of all measures will fall within ±2 standard deviations of the average. For example, if a set of data has an average of 10 and a standard deviation of 1, then 95% of the measures will be between 8 and 12. The normal distribution is a continuous curve that describes the probability for observing a particular value. The curve is normalized, meaning all data will be found within its bounds.

If you are making a prediction about future events or trying to understand a population based on a subset sample, then you calculate the confidence interval. The confidence interval quantifies the probability of observing a range of values. To calculate a confidence interval, you need to decide a degree of confidence, typically 95%, the standard deviation, and the number of observed values used to calculate the standard deviation. It is important to note that a distribution containing 95% of the observed data and a predicted 95% confidence interval (CI) are different. The confidence interval is making a prediction that the average is likely to occur anywhere within the bounds of the interval.

First, let's look at January and what you would have predicted for this month based on historic share prices. Averaging the closing price of past months of January, calculating the standard deviation and deriving a price channel of ±2 standard deviations, you would find that extrapolating a trend-line results in an average price of $531 with an upper bound of $587 and lower bound of $474. So far this month, the actual prices have fallen right in line with what would have been predicted a year ago. Looking at the year-over-year multiple, we see extremely robust price increases from 2005 to 2006. Using a Q test, we can eliminate the multiple of 4.5 as a data outlier. (Q = 0.545 and Q95% = 0.526 therefore Q > Q95%). After eliminating the outlier, the YoY average is 1.45 with standard deviation of 0.56 and a CI of ±0.45. Using the YoY multiple to predict this year's average share price, at 95% confidence, we would have expected a range from $432-$816. Based on these results, all of the current chaos actually looks very well behaved. With confidence, we would predict YoY growth for the month of January, though anemic growth is within the bounds of the interval.

January | 2005 | 2006 | 2007 | 2008 | 2009 | 2010 | 2011 | 2012 | Predicted 2013 |

Avg Close | $17.31 | $77.81 | $88.88 | $160.38 | $88.78 | $207.77 | $338.38 | $428.58 | $531.41 |

Std Dev (Sigma) | $0.91 | $4.28 | $4.45 | $21.39 | $4.27 | $5.62 | $5.64 | $13.00 | |

Lower 2Sigma | $15.49 | $69.25 | $79.99 | $117.59 | $80.24 | $196.53 | $327.10 | $402.57 | $474.94 |

Upper 2Sigma | $19.12 | $86.37 | $97.78 | $203.17 | $97.31 | $219.02 | $349.66 | $454.59 | $587.87 |

Year Over Year Multiple | 4.50 | 1.14 | 1.80 | 0.55 | 2.34 | 1.63 | 1.27 | 1.24 |

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Now let's skip forward to March. March tends to show more promise than does January. However, when we analyze the growth from January to March, we will find that much of what happens in March is baked in by January. What happens in the last week of January will matter. Again, averaging the closing price of past March prices, calculating the standard deviation and deriving a price channel of ±2 standard deviations, you would find that extrapolating a trend-line result in an average price of $703 with an upper bound of $763 and lower bound of $642. The big difference between March and January is the YoY multiple. March shows robust growth with confidence. The average YoY multiple is 1.52 with a standard deviation of 0.42 and a 95% CI of ±0.31. Based on the YoY multiple, at 95% confidence, March is predicted to have an average price fall between $653 and $1,102. Wow!

March | 2005 | 2006 | 2007 | 2008 | 2009 | 2010 | 2011 | 2012 | Predicted 2013 |

Avg Close | $42.00 | $64.04 | $90.60 | $130.82 | $97.51 | $223.41 | $347.49 | $577.51 | 702.50 |

Std Dev (Sigma) | $1.34 | $2.93 | $3.00 | $8.14 | $8.04 | $7.63 | $8.13 | $30.18 | |

Lower 2Sigma | $39.31 | $58.18 | $84.60 | $114.54 | $81.42 | $208.16 | $331.24 | $517.15 | $642.08 |

Upper 2Sigma | $44.68 | $69.89 | $96.61 | $147.10 | $113.59 | $238.66 | $363.74 | $637.87 | $762.94 |

Year Over Year Multiple | 1.52 | 1.41 | 1.44 | 0.75 | 2.29 | 1.56 | 1.66 | 1.22 |

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But wait, there's more, and this is when stuff gets really interesting. Let's look at the growth transition from January to March and see what we can learn. The below table shows the difference in YoY multiple between March and January i.e. March 2006 (64.04) divided by January 2006 (77.81) = 0.82. What we can see from the change in YoY multiples is that sometimes we get growth from January to March and sometimes we get contraction. The average change in January to March YoY multiple is 1.03, with a standard deviation of 0.17 and a 95% CI of ± 0.12. From this, we can't say with any confidence that there will be growth between the months of January and March. And, that is why I previously claimed that what happens in March is baked in January. Looking at the graph for the change multiple from January to March, we see that there appears to be a slight trend upward. The trend-line pegs this year's multiple at 1.23. Now when you multiply this year's January prediction of $531 by 1.23 you derive $652, which just so happens to fall within what you would see (within $1) YoY for March with confidence, and by trend-line. So there you have it, two ways, both of which point to $650 in March.

Jan to March | 2006 | 2007 | 2008 | 2009 | 2010 | 2011 | 2012 |

Growth Multiple | 0.82 | 1.02 | 0.82 | 1.10 | 1.08 | 1.03 | 1.35 |

*(click to enlarge)* *(click to enlarge)*

Lastly, onward to earnings. We have some key announcements from Apple that could provide good insight into what is coming during the January 23 earnings report. On 6/11, Apple reported greater than 400M App Store users. On 9/19, it was 435M and on 1/7/2013 it was over 500M. To be conservative, I'm going to redefine terms like "greater than" and "over" to mean "exactly." Given these dates between June and September, the App Store was adding users at a rate of 360,000 per day. From September to January, the company added users at a rate of 600,000 per day. Now, let's assume there is some relationship between new iPhone and iPad users with new App Store accounts. It seems reasonable that a new customer for an iPhone would be a new App Store user. Also, if someone is buying a second device, it might be reasonable to assume that the user wouldn't get a new account under those circumstances.

Looking at the previous earnings release, we see that Apple sold 41M iDevices (iPhones and iPads). That would represent sales of 500,000 iDevices per day for Q4 2012. Assuming that the relationship between iDevices and App Store users stays relatively constant, we can do some dimensional analysis and derive Q1 2013 iDevices sales. And, if proportions hold, then we can also derive sales and profit. So let's do it. iDevices per day divided by App Store users per day yields iDevices per App Store users. When we crunch the numbers for Q4 2012, we get 1.3 iDevices per App Store user. Paying that value forward yields 770,000 iDevices per day for a total of 69M iDevices for Q1 2013. Extrapolating that result to dollars produces sales of $58B and $14B in profit.

For the upside surprise, we have to assume when Apple announces that it had over 500M App Store accounts, the huge majority of those accounts were added before Christmas. With that assumption, we can back up some sales. Then what we find is that Q1 2013 delivers 83M iDevices with revenue of $70B and profit of $17B. And, there you have it. The consensus numbers are conservative and easily derived. Let's hope that Apple blossoms in the spring and some of the more robust numbers become a reality.

**Additional disclosure:** I'm a postdoctoral fellow at the FDA and serve under the quality management team in the office of policy and regulatory science.

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