Averages and spread

orders = [20, 25, 30, 25, 400]
print(sum(orders) / len(orders))

The mean: share it out equally

The mean is what most people call "the average": add everything up and divide by how many there are. Think of it as sharing out equally. If five friends' restaurant bills add up to $100, the mean is what each would pay if they split it evenly: $20.

bills = [18, 25, 12, 30, 15]
mean = sum(bills) / len(bills)
print(mean)

The mean has a useful property: the mean times the count gives back the total. That makes it the right choice when you care about totals, like revenue per customer to plan next month's revenue. Its weakness is that every value pulls on it, and extreme values pull hardest. One $400 order among $25 ones moves the mean a long way, as the opening question showed. And an empty list has no mean at all: sum([]) / len([]) divides by zero, so code that averages data it didn't check first can crash on a quiet day.

The median: the middle value

The median is the middle value once the list is sorted. Half the values are below it and half above. It doesn't care how extreme the extremes are: make the biggest order $400 or $4 million, and the median doesn't move.

times = [12, 3, 7, 30, 5]
in_order = sorted(times)
print(in_order)
middle = len(in_order) // 2
print(in_order[middle])

With an odd count there's one middle value: for five values it's at position 2, which is len // 2. With an even count there are two middle values, and the median is the mean of those two. That's why house prices and salaries are usually reported as medians: a few mansions or one very rich founder can't distort them.

times = sorted([12, 3, 7, 30, 5])
print(times[len(times) // 2])

hours = [34, 12, 27, 19, 45, 22]

in_order = hours    # sort it
median = 0          # the mean of the two middle values

print(in_order, median)

Percentiles: the median's family

The median is one of a family called percentiles. The 90th percentile is the value that 90% of the data falls below; the median is the 50th. Percentiles describe the slow end of response times well: "half our replies arrive within 2 hours, and 95% within 6" says far more than a single average.

Mean or median?

Neither is "the right average"; they answer different questions. The mean answers "if it were shared out equally, how much each?", which is what you need for totals and budgets. The median answers "what's a typical one like?", which is what you need when a few extreme values would mislead.

When the two are close, the data is fairly balanced. When the mean is far above the median, a few large values are pulling it up; that's called a skewed distribution, and it's common in money, waiting times and website traffic. Reporting both, side by side, tells a reader more than either alone. (A third average, the mode, is the most common value. It's handy for categories, like the most popular shirt size.)

Spread: how far from the middle?

Two shops both average 40 minutes to deliver. At the first, every delivery takes 38 to 42 minutes. At the second, some take 10 and some take 70. Same mean, very different experience. The spread says how far values typically sit from the middle.

The simplest measure is the range, max minus min, but one odd value can make it huge. The usual measure is the standard deviation: roughly, the typical distance from the mean. You find each value's distance from the mean, square the distances (so negatives don't cancel), average the squares, and take the square root to get back to the original units:

values = [38, 40, 42, 40]
mean = sum(values) / len(values)
total = 0
for v in values:
    total = total + (v - mean) ** 2
std = (total / len(values)) ** 0.5
print(mean, round(std, 2))

The average of the squared distances, before the square root, is called the variance. (Statisticians often divide by one less than the count when the list is a sample from a bigger group; the idea is the same.)

scores = [4, 8, 6, 5, 7]

mean = 0
variance = 0
std = 0

print(mean, variance, std)

Reading spread in results

For data shaped like a bell, a handy rule: about two thirds of the values sit within one standard deviation of the mean, and about 95% within two. So a report like "delivery takes 40 ± 5 minutes" says most deliveries take 35 to 45 minutes, and almost all take 30 to 50. A value three standard deviations out is rare enough to investigate.

The same notation shows up in AI results: "82% ± 3" means a model scored 82% on average across several runs, with a standard deviation of 3 points. If a new model scores 84% ± 3, the gap is smaller than the run-to-run wobble, and you can't yet claim it's better.

monday = 41 / 50 * 100
tuesday = 45 / 50 * 100
print((monday + tuesday) / 2)

Averages in AI

These three numbers are everywhere in the course. An eval's accuracy is a mean of 1s and 0s. A model's loss is a mean of its errors over a batch of examples. Results are reported as mean ± standard deviation across runs. Response times are reported as a median, plus a slow percentile such as the 95th, because a mean hides the slow tail. And in Phase 4, normalization layers subtract a mean and divide by a standard deviation inside the network itself.

Python's statistics module has mean(), median() and stdev() built in, and you'll meet it in the modules lesson. Having built each one by hand, you'll know exactly what they compute, and when one of them is the wrong number for the question.

scores = [0.8, 0.9, 0.7]
print(round(sum(scores) / len(scores), 2))