## 问题：熊猫轴是什么意思？

``````import pandas as pd
import numpy as np

dff = pd.DataFrame(np.random.randn(1,2),columns=list('AB'))
``````

``````+------------+---------+--------+
|            |  A      |  B     |
+------------+---------+---------
|      0     | 0.626386| 1.52325|
+------------+---------+--------+
``````

``dff.mean(axis=1)``

``````0    1.074821
dtype: float64
``````

``````A    0.626386
B    1.523255
dtype: float64
``````

Here is my code to generate a dataframe:

``````import pandas as pd
import numpy as np

dff = pd.DataFrame(np.random.randn(1,2),columns=list('AB'))
``````

then I got the dataframe:

``````+------------+---------+--------+
|            |  A      |  B     |
+------------+---------+---------
|      0     | 0.626386| 1.52325|
+------------+---------+--------+
``````

When I type the commmand :

``````dff.mean(axis=1)
``````

I got :

``````0    1.074821
dtype: float64
``````

According to the reference of pandas, axis=1 stands for columns and I expect the result of the command to be

``````A    0.626386
B    1.523255
dtype: float64
``````

So here is my question: what does axis in pandas mean?

## 回答 0

``````+------------+---------+--------+
|            |  A      |  B     |
+------------+---------+---------
|      0     | 0.626386| 1.52325|----axis=1----->
+------------+---------+--------+
|         |
| axis=0  |
↓         ↓
``````

It specifies the axis along which the means are computed. By default `axis=0`. This is consistent with the `numpy.mean` usage when `axis` is specified explicitly (in `numpy.mean`, axis==None by default, which computes the mean value over the flattened array) , in which `axis=0` along the rows (namely, index in pandas), and `axis=1` along the columns. For added clarity, one may choose to specify `axis='index'` (instead of `axis=0`) or `axis='columns'` (instead of `axis=1`).

``````+------------+---------+--------+
|            |  A      |  B     |
+------------+---------+---------
|      0     | 0.626386| 1.52325|----axis=1----->
+------------+---------+--------+
|         |
| axis=0  |
↓         ↓
``````

## 回答 1

• 轴0将作用于每个COLUMN中的所有ROWS
• 轴1将作用于每个行中的所有列

These answers do help explain this, but it still isn’t perfectly intuitive for a non-programmer (i.e. someone like me who is learning Python for the first time in context of data science coursework). I still find using the terms “along” or “for each” wrt to rows and columns to be confusing.

What makes more sense to me is to say it this way:

• Axis 0 will act on all the ROWS in each COLUMN
• Axis 1 will act on all the COLUMNS in each ROW

So a mean on axis 0 will be the mean of all the rows in each column, and a mean on axis 1 will be a mean of all the columns in each row.

Ultimately this is saying the same thing as @zhangxaochen and @Michael, but in a way that is easier for me to internalize.

## 回答 2

1. axis = 0表示沿“索引”。这是逐行操作

1. axis = 1表示沿“列”。这是列操作。

Let’s visualize (you gonna remember always),

In Pandas:

1. axis=0 means along “indexes”. It’s a row-wise operation.

Suppose, to perform concat() operation on dataframe1 & dataframe2, we will take dataframe1 & take out 1st row from dataframe1 and place into the new DF, then we take out another row from dataframe1 and put into new DF, we repeat this process until we reach to the bottom of dataframe1. Then, we do the same process for dataframe2.

Basically, stacking dataframe2 on top of dataframe1 or vice a versa.

E.g making a pile of books on a table or floor

1. axis=1 means along “columns”. It’s a column-wise operation.

Suppose, to perform concat() operation on dataframe1 & dataframe2, we will take out the 1st complete column(a.k.a 1st series) of dataframe1 and place into new DF, then we take out the second column of dataframe1 and keep adjacent to it (sideways), we have to repeat this operation until all columns are finished. Then, we repeat the same process on dataframe2. Basically, stacking dataframe2 sideways.

E.g arranging books on a bookshelf.

More to it, since arrays are better representations to represent a nested n-dimensional structure compared to matrices! so below can help you more to visualize how axis plays an important role when you generalize to more than one dimension. Also, you can actually print/write/draw/visualize any n-dim array but, writing or visualizing the same in a matrix representation(3-dim) is impossible on a paper more than 3-dimensions.

## 回答 3

`axis`指向数组的维，在`pd.DataFrame`s 的情况下`axis=0`是向下的维，`axis=1`而向右的维。

``a = np.ones((3,5,7))``

`a`是3维的`ndarray`，即具有3个轴（“轴”是“轴”的复数）。的配置`a`看起来像3片面包，每片面包的尺寸为5 x 7。`a[0,:,:]`将引用第0个切片，`a[1,:,:]`将引用第1 个切片，依此类推。

`a.sum(axis=0)``sum()`沿的第0轴应用`a`。您将添加所有切片，最后得到一个形状的切片`(5,7)`

`a.sum(axis=0)` 相当于

``````b = np.zeros((5,7))
for i in range(5):
for j in range(7):
b[i,j] += a[:,i,j].sum()``````

`b`并且`a.sum(axis=0)`将两者看起来像这样

``````array([[ 3.,  3.,  3.,  3.,  3.,  3.,  3.],
[ 3.,  3.,  3.,  3.,  3.,  3.,  3.],
[ 3.,  3.,  3.,  3.,  3.,  3.,  3.],
[ 3.,  3.,  3.,  3.,  3.,  3.,  3.],
[ 3.,  3.,  3.,  3.,  3.,  3.,  3.]])``````

`axis` refers to the dimension of the array, in the case of `pd.DataFrame`s `axis=0` is the dimension that points downwards and `axis=1` the one that points to the right.

Example: Think of an `ndarray` with shape `(3,5,7)`.

``````a = np.ones((3,5,7))
``````

`a` is a 3 dimensional `ndarray`, i.e. it has 3 axes (“axes” is plural of “axis”). The configuration of `a` will look like 3 slices of bread where each slice is of dimension 5-by-7. `a[0,:,:]` will refer to the 0-th slice, `a[1,:,:]` will refer to the 1-st slice etc.

`a.sum(axis=0)` will apply `sum()` along the 0-th axis of `a`. You will add all the slices and end up with one slice of shape `(5,7)`.

`a.sum(axis=0)` is equivalent to

``````b = np.zeros((5,7))
for i in range(5):
for j in range(7):
b[i,j] += a[:,i,j].sum()
``````

`b` and `a.sum(axis=0)` will both look like this

``````array([[ 3.,  3.,  3.,  3.,  3.,  3.,  3.],
[ 3.,  3.,  3.,  3.,  3.,  3.,  3.],
[ 3.,  3.,  3.,  3.,  3.,  3.,  3.],
[ 3.,  3.,  3.,  3.,  3.,  3.,  3.],
[ 3.,  3.,  3.,  3.,  3.,  3.,  3.]])
``````

In a `pd.DataFrame`, axes work the same way as in `numpy.array`s: `axis=0` will apply `sum()` or any other reduction function for each column.

N.B. In @zhangxaochen’s answer, I find the phrases “along the rows” and “along the columns” slightly confusing. `axis=0` should refer to “along each column”, and `axis=1` “along each row”.

## 回答 4

`axis = 0`：按列=按列=沿行

`axis = 1`：按行=按行=沿列

The easiest way for me to understand is to talk about whether you are calculating a statistic for each column (`axis = 0`) or each row (`axis = 1`). If you calculate a statistic, say a mean, with `axis = 0` you will get that statistic for each column. So if each observation is a row and each variable is in a column, you would get the mean of each variable. If you set `axis = 1` then you will calculate your statistic for each row. In our example, you would get the mean for each observation across all of your variables (perhaps you want the average of related measures).

`axis = 0`: by column = column-wise = along the rows

`axis = 1`: by row = row-wise = along the columns

## 回答 5

1.第1轴将对所有列的每一行起作用

2.轴0将对所有行的每一列起作用

Let’s look at the table from Wiki. This is an IMF estimate of GDP from 2010 to 2019 for top ten countries.

1. Axis 1 will act for each row on all the columns
If you want to calculate the average (mean) GDP for EACH countries over the decade (2010-2019), you need to do, `df.mean(axis=1)`. For example, if you want to calculate mean GDP of United States from 2010 to 2019, `df.loc['United States','2010':'2019'].mean(axis=1)`

2. Axis 0 will act for each column on all the rows
If I want to calculate the average (mean) GDP for EACH year for all countries, you need to do, `df.mean(axis=0)`. For example, if you want to calculate mean GDP of the year 2015 for United States, China, Japan, Germany and India, `df.loc['United States':'India','2015'].mean(axis=0)`

Note: The above code will work only after setting “Country(or dependent territory)” column as the Index, using `set_index` method.

## 回答 6

``````import numpy as np

a=np.arange(120).reshape(2,3,4,5)

a.shape
Out[3]: (2, 3, 4, 5)

np.sum(a,axis=0).shape
Out[4]: (3, 4, 5)

np.sum(a,axis=1).shape
Out[5]: (2, 4, 5)

np.sum(a,axis=2).shape
Out[6]: (2, 3, 5)

np.sum(a,axis=3).shape
Out[7]: (2, 3, 4)``````

Axis in view of programming is the position in the shape tuple. Here is an example:

``````import numpy as np

a=np.arange(120).reshape(2,3,4,5)

a.shape
Out[3]: (2, 3, 4, 5)

np.sum(a,axis=0).shape
Out[4]: (3, 4, 5)

np.sum(a,axis=1).shape
Out[5]: (2, 4, 5)

np.sum(a,axis=2).shape
Out[6]: (2, 3, 5)

np.sum(a,axis=3).shape
Out[7]: (2, 3, 4)
``````

Mean on the axis will cause that dimension to be removed.

Referring to the original question, the dff shape is (1,2). Using axis=1 will change the shape to (1,).

## 回答 7

The designer of pandas, Wes McKinney, used to work intensively on finance data. Think of columns as stock names and index as daily prices. You can then guess what the default behavior is (i.e., `axis=0`) with respect to this finance data. `axis=1` can be simply thought as ‘the other direction’.

For example, the statistics functions, such as `mean()`, `sum()`, `describe()`, `count()` all default to column-wise because it makes more sense to do them for each stock. `sort_index(by=)` also defaults to column. `fillna(method='ffill')` will fill along column because it is the same stock. `dropna()` defaults to row because you probably just want to discard the price on that day instead of throw away all prices of that stock.

Similarly, the square brackets indexing refers to the columns since it’s more common to pick a stock instead of picking a day.

## 回答 8

• 如果您期望每行的输出都使用axis =’columns’，
• 另一方面，如果要为每列输出，请使用axis =’rows’。

one of easy ways to remember axis 1 (columns), vs axis 0 (rows) is the output you expect.

• if you expect an output for each row you use axis=’columns’,
• on the other hand if you want an output for each column you use axis=’rows’.

## 回答 9

`axis=`正确使用的问题是在两种主要情况下的使用：

1. 用于计算累积值重新排列（例如排序）数据。
2. 用于操纵（“播放”）实体（例如dataframe）。

• `0` 对于x轴，
• `1` y轴
• `2` 对于z轴。

z轴是只对面板 ; 对于数据帧，我们将把兴趣限制在带有x轴（，垂直）y轴（，水平）的绿色二维基本平面`0``1`

1. 如果要计算累加值，则可以从沿轴0（或沿轴1）定位的值（使用`axis=0``axis=1`）计算得出。

同样，如果要重新排列值，请使用轴的轴编号沿着该轴的编号将放置数据以进行重新排列（例如，用于排序）。

2. 如果要操作（例如连接实体（例如，数据框），请使用`axis='index'`（同义词：）`axis='rows'``axis='columns'`指定结果更改 – 分别为索引）或
（对于串联，您将分别获得更长的索引（=更多的行）更多的列。）

The problem with using `axis=` properly is for its use for 2 main different cases:

1. For computing an accumulated value, or rearranging (e. g. sorting) data.
2. For manipulating (“playing” with) entities (e. g. dataframes).

The main idea behind this answer is that for avoiding the confusion, we select either a number, or a name for specifying the particular axis, whichever is more clear, intuitive, and descriptive.

Pandas is based on NumPy, which is based on mathematics, particularly on n-dimensional matrices. Here is an image for common use of axes’ names in math in the 3-dimensional space:

This picture is for memorizing the axes’ ordinal numbers only:

• `0` for x-axis,
• `1` for y-axis, and
• `2` for z-axis.

The z-axis is only for panels; for dataframes we will restrict our interest to the green-colored, 2-dimensional basic plane with x-axis (`0`, vertical), and y-axis (`1`, horizontal).

It’s all for numbers as potential values of `axis=` parameter.

The names of axes are `'index'` (you may use the alias `'rows'`) and `'columns'`, and for this explanation it is NOT important the relation between these names and ordinal numbers (of axes), as everybody knows what the words “rows” and “columns” mean (and everybody here — I suppose — knows what the word “index” in pandas means).

And now, my recommendation:

1. If you want to compute an accumulated value, you may compute it from values located along axis 0 (or along axis 1) — use `axis=0` (or `axis=1`).

Similarly, if you want to rearrange values, use the axis number of the axis, along which are located data for rearranging (e.g. for sorting).

2. If you want to manipulate (e.g. concatenate) entities (e.g. dataframes) — use `axis='index'` (synonym: `axis='rows'`) or `axis='columns'` to specify the resulting changeindex (rows) or columns, respectively.
(For concatenating, you will obtain either a longer index (= more rows), or more columns, respectively.)

## 回答 10

`` a = np.ones((3,5,7))``

``````    array([[[1., 1., 1., 1., 1., 1., 1.],
[1., 1., 1., 1., 1., 1., 1.],
[1., 1., 1., 1., 1., 1., 1.],
[1., 1., 1., 1., 1., 1., 1.],
[1., 1., 1., 1., 1., 1., 1.]],

[[1., 1., 1., 1., 1., 1., 1.],
[1., 1., 1., 1., 1., 1., 1.],
[1., 1., 1., 1., 1., 1., 1.],
[1., 1., 1., 1., 1., 1., 1.],
[1., 1., 1., 1., 1., 1., 1.]],

[[1., 1., 1., 1., 1., 1., 1.],
[1., 1., 1., 1., 1., 1., 1.],
[1., 1., 1., 1., 1., 1., 1.],
[1., 1., 1., 1., 1., 1., 1.],
[1., 1., 1., 1., 1., 1., 1.]]])``````

`````` x0 = np.sum(a,axis=0)
x1 = np.sum(a,axis=1)
x2 = np.sum(a,axis=2)``````

``````   x0 :
array([[3., 3., 3., 3., 3., 3., 3.],
[3., 3., 3., 3., 3., 3., 3.],
[3., 3., 3., 3., 3., 3., 3.],
[3., 3., 3., 3., 3., 3., 3.],
[3., 3., 3., 3., 3., 3., 3.]])

x1 :
array([[5., 5., 5., 5., 5., 5., 5.],
[5., 5., 5., 5., 5., 5., 5.],
[5., 5., 5., 5., 5., 5., 5.]])

x2 :
array([[7., 7., 7., 7., 7.],
[7., 7., 7., 7., 7.],
[7., 7., 7., 7., 7.]])``````

This is based on @Safak’s answer. The best way to understand the axes in pandas/numpy is to create a 3d array and check the result of the sum function along the 3 different axes.

`````` a = np.ones((3,5,7))
``````

a will be:

``````    array([[[1., 1., 1., 1., 1., 1., 1.],
[1., 1., 1., 1., 1., 1., 1.],
[1., 1., 1., 1., 1., 1., 1.],
[1., 1., 1., 1., 1., 1., 1.],
[1., 1., 1., 1., 1., 1., 1.]],

[[1., 1., 1., 1., 1., 1., 1.],
[1., 1., 1., 1., 1., 1., 1.],
[1., 1., 1., 1., 1., 1., 1.],
[1., 1., 1., 1., 1., 1., 1.],
[1., 1., 1., 1., 1., 1., 1.]],

[[1., 1., 1., 1., 1., 1., 1.],
[1., 1., 1., 1., 1., 1., 1.],
[1., 1., 1., 1., 1., 1., 1.],
[1., 1., 1., 1., 1., 1., 1.],
[1., 1., 1., 1., 1., 1., 1.]]])
``````

Now check out the sum of elements of the array along each of the axes:

`````` x0 = np.sum(a,axis=0)
x1 = np.sum(a,axis=1)
x2 = np.sum(a,axis=2)
``````

will give you the following results:

``````   x0 :
array([[3., 3., 3., 3., 3., 3., 3.],
[3., 3., 3., 3., 3., 3., 3.],
[3., 3., 3., 3., 3., 3., 3.],
[3., 3., 3., 3., 3., 3., 3.],
[3., 3., 3., 3., 3., 3., 3.]])

x1 :
array([[5., 5., 5., 5., 5., 5., 5.],
[5., 5., 5., 5., 5., 5., 5.],
[5., 5., 5., 5., 5., 5., 5.]])

x2 :
array([[7., 7., 7., 7., 7.],
[7., 7., 7., 7., 7.],
[7., 7., 7., 7., 7.]])
``````

## 回答 11

``````df = pd.DataFrame(np.arange(12).reshape(3,4),columns=['A', 'B', 'C', 'D'])
print(df)
A  B   C   D
0  0  1   2   3
1  4  5   6   7
2  8  9  10  11

df.mean(axis=1)

0    1.5
1    5.5
2    9.5
dtype: float64

df.drop(['A','B'],axis=1,inplace=True)

C   D
0   2   3
1   6   7
2  10  11``````

I understand this way :

Say if your operation requires traversing from left to right/right to left in a dataframe, you are apparently merging columns ie. you are operating on various columns. This is axis =1

Example

``````df = pd.DataFrame(np.arange(12).reshape(3,4),columns=['A', 'B', 'C', 'D'])
print(df)
A  B   C   D
0  0  1   2   3
1  4  5   6   7
2  8  9  10  11

df.mean(axis=1)

0    1.5
1    5.5
2    9.5
dtype: float64

df.drop(['A','B'],axis=1,inplace=True)

C   D
0   2   3
1   6   7
2  10  11
``````

Point to note here is we are operating on columns

Similarly, if your operation requires traversing from top to bottom/bottom to top in a dataframe, you are merging rows. This is axis=0.

## 回答 12

``sums[key] = lang_sets[key].iloc[:,1:].sum(axis=0)``

axis = 0 means up to down axis = 1 means left to right

``````sums[key] = lang_sets[key].iloc[:,1:].sum(axis=0)
``````

Given example is taking sum of all the data in column == key.

## 回答 13

My thinking : Axis = n, where n = 0, 1, etc. means that the matrix is collapsed (folded) along that axis. So in a 2D matrix, when you collapse along 0 (rows), you are really operating on one column at a time. Similarly for higher order matrices.

This is not the same as the normal reference to a dimension in a matrix, where 0 -> row and 1 -> column. Similarly for other dimensions in an N dimension array.

## 回答 14

0列向下|

1行列向右->

I’m a newbie to pandas. But this is how I understand axis in pandas:

Axis Constant Varying Direction

0 Column Row Downwards |

1 Row Column Towards Right –>

So to compute mean of a column, that particular column should be constant but the rows under that can change (varying) so it is axis=0.

Similarly, to compute mean of a row, that particular row is constant but it can traverse through different columns (varying), axis=1.

## 回答 15

``````np.mean(np.array(np.ones(shape=(3,5,10))),axis = 0).shape # (5,10)
np.mean(np.array(np.ones(shape=(3,5,10))),axis = 1).shape # (3,10)
np.mean(np.array(np.ones(shape=(3,5,10))),axis = (0,1)).shape # (10,)``````

I think there is an another way to understand it.

For a np.array,if we want eliminate columns we use axis = 1; if we want eliminate rows, we use axis = 0.

``````np.mean(np.array(np.ones(shape=(3,5,10))),axis = 0).shape # (5,10)
np.mean(np.array(np.ones(shape=(3,5,10))),axis = 1).shape # (3,10)
np.mean(np.array(np.ones(shape=(3,5,10))),axis = (0,1)).shape # (10,)
``````

For pandas object, `axis = 0` stands for row-wise operation and `axis = 1` stands for column-wise operation. This is different from `numpy` by definition, we can check definitions from numpy.doc and pandas.doc

## 回答 16

0和1只是’row’和’column’的别名。这是矩阵索引的惯例。

I will explicitly avoid using ‘row-wise’ or ‘along the columns’, since people may interpret them in exactly the wrong way.

Analogy first. Intuitively, you would expect that `pandas.DataFrame.drop(axis='column')` drops a column from N columns and gives you (N – 1) columns. So you can pay NO attention to rows for now (and remove word ‘row’ from your English dictionary.) Vice versa, `drop(axis='row')` works on rows.

In the same way, `sum(axis='column')` works on multiple columns and gives you 1 column. Similarly, `sum(axis='row')` results in 1 row. This is consistent with its simplest form of definition, reducing a list of numbers to a single number.

In general, with `axis=column`, you see columns, work on columns, and get columns. Forget rows.

With `axis=row`, change perspective and work on rows.

0 and 1 are just aliases for ‘row’ and ‘column’. It’s the convention of matrix indexing.

## 回答 17

``````+------------+---------+--------+
|            |  A      |  B     |
+------------+---------+---------
|      X     | 0.626386| 1.52325|
+------------+---------+--------+
|      Y     | 0.626386| 1.52325|
+------------+---------+--------+``````

``````+------------+---------+--------+
|            |  A      |  B     |``````

``|            | 0.626386| 1.52325|  ``

``|            | 0.626386| 1.52325|``

``````+------------+
|      X     |
+------------+
|      Y     |
+------------+``````

``````+------------+---------+
|      X     | 0.626386
+------------+---------+
|      Y     | 0.626386
+------------+---------+``````

``````+------------+---------+
|      X     | 1.52325 |
+------------+---------+
|      Y     | 1.52325 |
+------------+---------+``````

I have been trying to figure out the axis for the last hour as well. The language in all the above answers, and also the documentation is not at all helpful.

To answer the question as I understand it now, in Pandas, axis = 1 or 0 means which axis headers do you want to keep constant when applying the function.

Note: When I say headers, I mean index names

``````+------------+---------+--------+
|            |  A      |  B     |
+------------+---------+---------
|      X     | 0.626386| 1.52325|
+------------+---------+--------+
|      Y     | 0.626386| 1.52325|
+------------+---------+--------+
``````

For axis=1=columns : We keep columns headers constant and apply the mean function by changing data. To demonstrate, we keep the columns headers constant as:

``````+------------+---------+--------+
|            |  A      |  B     |
``````

Now we populate one set of A and B values and then find the mean

``````|            | 0.626386| 1.52325|
``````

Then we populate next set of A and B values and find the mean

``````|            | 0.626386| 1.52325|
``````

Similarly, for axis=rows, we keep row headers constant, and keep changing the data: To demonstrate, first fix the row headers:

``````+------------+
|      X     |
+------------+
|      Y     |
+------------+
``````

Now populate first set of X and Y values and then find the mean

``````+------------+---------+
|      X     | 0.626386
+------------+---------+
|      Y     | 0.626386
+------------+---------+
``````

Then populate the next set of X and Y values and then find the mean:

``````+------------+---------+
|      X     | 1.52325 |
+------------+---------+
|      Y     | 1.52325 |
+------------+---------+
``````

In summary,

When axis=columns, you fix the column headers and change data, which will come from the different rows.

When axis=rows, you fix the row headers and change data, which will come from the different columns.

## 回答 18

axis = 1，它将明智地求和行，keepdims = True将保持二维。希望对您有帮助。

axis=1 ,It will give the sum row wise,keepdims=True will maintain the 2D dimension. Hope it helps you.

## 回答 19

• 有趣的是，与二维数组相比，使用三维数组更容易理解它们的行为。
• 在Python包中 `numpy``pandas`，sum的axis参数实际上指定numpy，以计算所有可以以array [0，0，…，i，…，0]形式获取的值的平均值所有可能的值。在i的位置固定的情况下重复此过程，其他维度的索引则一个接一个地变化（从最右边的元素开始）。结果是一个n-1维数组。
• 在R中，MARGINS参数使`apply`函数计算可以以array [，…，i，…，]的形式获取的所有值的平均值，其中i遍历所有可能的值。迭代完所有i值后，不再重复该过程。因此，结果是一个简单的向量。

Many answers here helped me a lot!

In case you get confused by the different behaviours of `axis` in Python and `MARGIN` in R (like in the `apply` function), you may find a blog post that I wrote of interest: https://accio.github.io/programming/2020/05/19/numpy-pandas-axis.html.

In essence:

• Their behaviours are, intriguingly, easier to understand with three-dimensional array than with two-dimensional arrays.
• In Python packages `numpy` and `pandas`, the axis parameter in sum actually specifies numpy to calculate the mean of all values that can be fetched in the form of array[0, 0, …, i, …, 0] where i iterates through all possible values. The process is repeated with the position of i fixed and the indices of other dimensions vary one after the other (from the most far-right element). The result is a n-1-dimensional array.
• In R, the MARGINS parameter let the `apply` function calculate the mean of all values that can be fetched in the form of array[, … , i, … ,] where i iterates through all possible values. The process is not repeated when all i values have been iterated. Therefore, the result is a simple vector.

## 回答 20

Arrays are designed with so-called axis=0 and rows positioned vertically versus axis=1 and columns positioned horizontally. Axis refers to the dimension of the array.