[Fix](variance) Fix sample variance/stddev null res for single value

docs/sql-manual/sql-functions/aggregate-functions/stddev-samp.md

@@ -10,6 +10,14 @@
 
 Returns the sample standard deviation of the expr expression
 
+The calculation formula is:
+
+$
+\mathrm{STDDEV\_SAMP}(x)=\sqrt{\mathrm{VAR\_SAMP}(x)}=\sqrt{\frac{\sum_{i=1}^{n}(x_i-\bar{x})^2}{n-1}}
+$
+
+Where `n` is the number of valid values in the group.
+
 ## Syntax
 
 ```sql
@@ -25,11 +33,11 @@ STDDEV_SAMP(<expr>)
 ## Return Value
 
 Return the sample standard deviation of the expr expression as Double type.
-If there is no valid data in the group, returns NULL.
+If there is no valid data in the group, returns NULL. If the number of valid values in the group is 1, returns NaN.
 
 ### Examples
 ```sql
--- Create sample tables
+-- Create sample table
 CREATE TABLE score_table (
     student_id INT,
     score DOUBLE
@@ -58,3 +66,24 @@ FROM score_table;
 | 4.949747468305831 |
 +-------------------+
 ```
+
+When the number of valid values is 1, `STDDEV_SAMP` returns `NaN`.
+
+```sql
+-- Create a single-column sample table
+CREATE TABLE sample_values (
+    value INT
+) DISTRIBUTED BY HASH(value)
+PROPERTIES (
+    "replication_num" = "1"
+);
+
+INSERT INTO sample_values VALUES (10);
+
+SELECT STDDEV_SAMP(value) AS sample_stddev FROM sample_values;
++---------------+
+| sample_stddev |
++---------------+
+|           NaN |
++---------------+
+```

docs/sql-manual/sql-functions/aggregate-functions/var-samp.md

@@ -10,6 +10,14 @@
 
 The VAR_SAMP function calculates the sample variance of a specified expression. Unlike VARIANCE (population variance), VAR_SAMP uses n-1 as the divisor, which is considered an unbiased estimate of the population variance in statistics.
 
+The calculation formula is:
+
+$
+\mathrm{VAR\_SAMP}(x)=\frac{\sum_{i=1}^{n}(x_i-\bar{x})^2}{n-1}
+$
+
+Where `n` is the number of valid values in the group.
+
 ## Alias
 
 - VARIANCE_SAMP
@@ -28,7 +36,7 @@ VAR_SAMP(<expr>)
 
 ## Return Value
 Returns a Double value representing the calculated sample variance.
-If there is no valid data in the group, returns NULL.
+If there is no valid data in the group, returns NULL. If the number of valid values in the group is 1, returns NaN.
 
 ## Examples
 ```sql
@@ -66,3 +74,24 @@ FROM student_scores;
 | 29.4107142857143 |   25.73437500000001 |
 +------------------+---------------------+
 ```
+
+When the number of valid values is 1, `VAR_SAMP` returns `NaN`.
+
+```sql
+-- Create a single-column sample table
+CREATE TABLE sample_values (
+    value INT
+) DISTRIBUTED BY HASH(value)
+PROPERTIES (
+    "replication_num" = "1"
+);
+
+INSERT INTO sample_values VALUES (10);
+
+SELECT VAR_SAMP(value) AS sample_variance FROM sample_values;
++-----------------+
+| sample_variance |
++-----------------+
+|             NaN |
++-----------------+
+```

i18n/zh-CN/docusaurus-plugin-content-docs/current/sql-manual/sql-functions/aggregate-functions/stddev-samp.md

@@ -10,6 +10,14 @@
 
 返回 expr 表达式的样本标准差
 
+计算公式如下：
+
+$
+\mathrm{STDDEV\_SAMP}(x)=\sqrt{\mathrm{VAR\_SAMP}(x)}=\sqrt{\frac{\sum_{i=1}^{n}(x_i-\bar{x})^2}{n-1}}
+$
+
+其中 `n` 为组内合法数据的个数。
+
 ## 语法
 
 ```sql
@@ -25,7 +33,7 @@ STDDEV_SAMP(<expr>)
 ## 返回值
 
 返回 Double 类型的参数 expr 的样本标准差。
-当组内没有合法数据时，返回 NULL。
+当组内没有合法数据时，返回 NULL。组内合法数据个数为 1 时，返回 NaN。
 
 ## 举例
 ```sql
@@ -58,3 +66,24 @@ FROM score_table;
 | 4.949747468305831 |
 +-------------------+
 ```
+
+当合法数据个数为 1 时，`STDDEV_SAMP` 返回 `NaN`。
+
+```sql
+-- 创建单列示例表
+CREATE TABLE sample_values (
+    value INT
+) DISTRIBUTED BY HASH(value)
+PROPERTIES (
+    "replication_num" = "1"
+);
+
+INSERT INTO sample_values VALUES (10);
+
+SELECT STDDEV_SAMP(value) AS sample_stddev FROM sample_values;
++---------------+
+| sample_stddev |
++---------------+
+|           NaN |
++---------------+
+```

i18n/zh-CN/docusaurus-plugin-content-docs/current/sql-manual/sql-functions/aggregate-functions/var-samp.md

@@ -10,6 +10,14 @@
 
 VAR_SAMP 函数计算指定表达式的样本方差。与 VARIANCE（总体方差）不同，VAR_SAMP 使用 n-1 作为除数，这在统计学上被认为是对总体方差的无偏估计。
 
+计算公式如下：
+
+$
+\mathrm{VAR\_SAMP}(x)=\frac{\sum_{i=1}^{n}(x_i-\bar{x})^2}{n-1}
+$
+
+其中 `n` 为组内合法数据的个数。
+
 ## 别名
 
 - VARIANCE_SAMP
@@ -28,7 +36,7 @@ VAR_SAMP(<expr>)
 
 ## 返回值
 返回一个 Double 类型的值，表示计算得到的样本方差。
-组内没有合法数据时，返回 NULL。
+组内没有合法数据时，返回 NULL。组内合法数据个数为 1 时，返回 NaN。
 
 ## 举例
 ```sql
@@ -66,3 +74,24 @@ FROM student_scores;
 | 29.4107142857143 |   25.73437500000001 |
 +------------------+---------------------+
 ```
+
+当合法数据个数为 1 时，`VAR_SAMP` 返回 `NaN`。
+
+```sql
+-- 创建单列示例表
+CREATE TABLE sample_values (
+    value INT
+) DISTRIBUTED BY HASH(value)
+PROPERTIES (
+    "replication_num" = "1"
+);
+
+INSERT INTO sample_values VALUES (10);
+
+SELECT VAR_SAMP(value) AS sample_variance FROM sample_values;
++-----------------+
+| sample_variance |
++-----------------+
+|             NaN |
++-----------------+
+```

i18n/zh-CN/docusaurus-plugin-content-docs/version-4.x/sql-manual/sql-functions/aggregate-functions/stddev-samp.md

@@ -1,4 +1,4 @@
 ---
 {
     "title": "STDDEV_SAMP",
     "language": "zh-CN",
@@ -10,6 +10,14 @@
 
 返回 expr 表达式的样本标准差
 
+计算公式如下：
+
+$
+\mathrm{STDDEV\_SAMP}(x)=\sqrt{\mathrm{VAR\_SAMP}(x)}=\sqrt{\frac{\sum_{i=1}^{n}(x_i-\bar{x})^2}{n-1}}
+$
+
+其中 `n` 为组内合法数据的个数。
+
 ## 语法
 
 ```sql
@@ -25,7 +33,7 @@
 ## 返回值
 
 返回 Double 类型的参数 expr 的样本标准差。
-当组内没有合法数据时，返回 NULL。
+当组内没有合法数据时，返回 NULL。组内合法数据个数为 1 时，返回 NaN。
 
 ## 举例
 ```sql
@@ -58,3 +66,24 @@
 | 4.949747468305831 |
 +-------------------+
 ```
+
+当合法数据个数为 1 时，`STDDEV_SAMP` 返回 `NaN`。
+
+```sql
+-- 创建单列示例表
+CREATE TABLE sample_values (
+    value INT
+) DISTRIBUTED BY HASH(value)
+PROPERTIES (
+    "replication_num" = "1"
+);
+
+INSERT INTO sample_values VALUES (10);
+
+SELECT STDDEV_SAMP(value) AS sample_stddev FROM sample_values;
++---------------+
+| sample_stddev |
++---------------+
+|           NaN |
++---------------+
+```

i18n/zh-CN/docusaurus-plugin-content-docs/version-4.x/sql-manual/sql-functions/aggregate-functions/var-samp.md

@@ -10,6 +10,14 @@
 
 VAR_SAMP 函数计算指定表达式的样本方差。与 VARIANCE（总体方差）不同，VAR_SAMP 使用 n-1 作为除数，这在统计学上被认为是对总体方差的无偏估计。
 
+计算公式如下：
+
+$
+\mathrm{VAR\_SAMP}(x)=\frac{\sum_{i=1}^{n}(x_i-\bar{x})^2}{n-1}
+$
+
+其中 `n` 为组内合法数据的个数。
+
 ## 别名
 
 - VARIANCE_SAMP
@@ -28,7 +36,7 @@ VAR_SAMP(<expr>)
 
 ## 返回值
 返回一个 Double 类型的值，表示计算得到的样本方差。
-组内没有合法数据时，返回 NULL。
+组内没有合法数据时，返回 NULL。组内合法数据个数为 1 时，返回 NaN。
 
 ## 举例
 ```sql
@@ -66,3 +74,24 @@ FROM student_scores;
 | 29.4107142857143 |   25.73437500000001 |
 +------------------+---------------------+
 ```
+
+当合法数据个数为 1 时，`VAR_SAMP` 返回 `NaN`。
+
+```sql
+-- 创建单列示例表
+CREATE TABLE sample_values (
+    value INT
+) DISTRIBUTED BY HASH(value)
+PROPERTIES (
+    "replication_num" = "1"
+);
+
+INSERT INTO sample_values VALUES (10);
+
+SELECT VAR_SAMP(value) AS sample_variance FROM sample_values;
++-----------------+
+| sample_variance |
++-----------------+
+|             NaN |
++-----------------+
+```

versioned_docs/version-4.x/sql-manual/sql-functions/aggregate-functions/stddev-samp.md

@@ -1,4 +1,4 @@
 ---
 {
     "title": "STDDEV_SAMP",
     "language": "en",
@@ -10,6 +10,14 @@
 
 Returns the sample standard deviation of the expr expression
 
+The calculation formula is:
+
+$
+\mathrm{STDDEV\_SAMP}(x)=\sqrt{\mathrm{VAR\_SAMP}(x)}=\sqrt{\frac{\sum_{i=1}^{n}(x_i-\bar{x})^2}{n-1}}
+$
+
+Where `n` is the number of valid values in the group.
+
 ## Syntax
 
 ```sql
@@ -25,11 +33,11 @@
 ## Return Value
 
 Return the sample standard deviation of the expr expression as Double type.
-If there is no valid data in the group, returns NULL.
+If there is no valid data in the group, returns NULL. If the number of valid values in the group is 1, returns NaN.
 
 ### Examples
 ```sql
--- Create sample tables
+-- Create sample table
 CREATE TABLE score_table (
     student_id INT,
     score DOUBLE
@@ -58,3 +66,24 @@
 | 4.949747468305831 |
 +-------------------+
 ```
+
+When the number of valid values is 1, `STDDEV_SAMP` returns `NaN`.
+
+```sql
+-- Create a single-column sample table
+CREATE TABLE sample_values (
+    value INT
+) DISTRIBUTED BY HASH(value)
+PROPERTIES (
+    "replication_num" = "1"
+);
+
+INSERT INTO sample_values VALUES (10);
+
+SELECT STDDEV_SAMP(value) AS sample_stddev FROM sample_values;
++---------------+
+| sample_stddev |
++---------------+
+|           NaN |
++---------------+
+```