Summaries and shapes

What you will be able to do

After this lesson you can

• find the sum, the mean, the spread and the places of the biggest and smallest numbers,
• work with a two-dimensional array, a table of numbers,
• find the average of every row or every column with axis, and
• change the shape of an array and use broadcasting.

One number from many

Predict first. Which place holds the tallest height? Then run it.

The heights of six friends are in centimetres.

• h.sum(), h.min() and h.max() give the total (923), the smallest (148) and the biggest (160).
• h.mean() is the average, 153.83. np.median(h) is the middle value, 153.5. (The median is a function of np, and not a method of the array.)
• h.std() is the standard deviation, a measure of how spread out the heights are. NumPy divides by $n$ by default. The statistics of a sample divide by $n - 1$, and you ask for this with ddof=1: 4.67 and not 4.26. When your numbers are a sample of a larger population, which is usual in an experiment, use ddof=1.
• h.argmax() is the place of the biggest number: 2 (the value 160). h.argmin() is the place of the smallest: 1 (the value 148). Use argmax when you want to know which, and max when you want to know how much.
• h.cumsum() is the running total: 152, 152 + 148 = 300, and so on.
• np.diff(h) is the difference between each number and the one before: [-4 12 -5 -5 8]. It has one number fewer than h.
• np.sort(h)[::-1][:3] sorts, reverses, and takes the first three: the three tallest.

Tables: two dimensions

Predict first. What is marks[1, 2]? Then run it.

A table of numbers is a two-dimensional array. The marks of three students (the rows) in four tests (the columns) have the shape (3, 4): 3 rows and 4 columns. marks.size is the total number of numbers: 12.

• marks[1, 2] is the number in row 1, column 2. Counting starts at 0, so it is the second student in the third test: 55. The row comes first, then the column.
• marks[0] is the whole first row. marks[:, 0] means "all the rows, column 0": the first column, the first test of all the students. The colon means "everything".
• marks[:2, 1:3] is a smaller table: rows 0 and 1, columns 1 and 2.

Now the important idea, the axis.

• marks.mean(axis=1) gives one average for each row (each student): it works along the columns of each row. The result is [75. 60. 88.5].
• marks.mean(axis=0) gives one average for each column (each test): it works down the rows. The result has four numbers.
• With no axis, the method uses all the numbers: marks.max() is 95, the best mark in the whole table.

A way to remember: axis=0 collapses the rows, leaving one number per column. axis=1 collapses the columns, leaving one number per row.

Changing the shape

The axis you name is the one that disappears: axis=0 adds over the rows, so only the columns are left.

Broadcasting

The rule in short: compare the shapes from the right. Two sizes fit if they are equal or one of them is 1 (or missing). Then the size 1 is stretched. If neither is the case, you get the ValueError that you saw in the last lesson.

Common mistakes

• Mixing up the axes. axis=0 gives one value per column, axis=1 one value per row.
• Writing marks[1][2] and marks[1, 2] as if they were different. For an array they give the same number, but marks[1, 2] is the usual way.
• Using the standard deviation of a population for a sample. Add ddof=1 for a sample.
• Reshaping to a shape that does not fit. The product of the sizes must be the number of numbers.
• Mixing up argmax and max. argmax is the place. max is the value.

Practice

Start with two quick questions. Each New numbers gives you a fresh one, so you can practise as long as you like.

Then the labs and the problems:

1. Warm-up. Open the lab Summarise a data set. You write a function that returns the main numbers of an array.
2. Build. Open the lab Marks of a class and the problem Rows and columns.
3. Stretch. Open the problem Standard deviation of a sample.

Remember these

• sum, min, max, mean, std, argmax, argmin, np.median, cumsum, np.diff.
• std() divides by $n$. For a sample, use std(ddof=1).
• A table is a two-dimensional array. a[row, column], a[:, 0] for a column.
• axis=0 gives one value per column. axis=1 gives one value per row.
• reshape, .T and broadcasting.

Check yourself

Try each one on your own first. If you are stuck, open a hint. Then check with Show the answer.

1. What is the shape of a table with 4 rows and 6 columns?

2. For t = np.array([[1, 2], [3, 4]]), what is t.sum(axis=0)?

3. What is t.sum(axis=1)?

4. Which number does np.array([4, 9, 2]).argmax() give?