There are other ways...
VARIANCE is STDDEV squared.
The two variances vary by a factor of COUNT/(count-1)
You don't have to compute the MEAN before starting the STDDEV; see the alternative formula used below, and in
http://en.wikipedia.org/wiki/Standard_deviation
SELECT
city,
COUNT(*),
AVG(temp),
ROUND(SUM(temp*temp)/COUNT(*) - AVG(temp)*AVG(temp), 2) AS Var1,
ROUND(COUNT(*) / (COUNT(*)-1) *
(SUM(temp*temp)/COUNT(*) - AVG(temp)*AVG(temp)), 2) AS Var_A,
STD(temp)*STD(temp) AS 'STD**2',
COUNT(*) / (COUNT(*)-1) *
STD(temp)*STD(temp) AS 'STD_A**2'
FROM temperatures
GROUP BY DATE(date), city;
+-----------+----------+-----------+------+-------+--------+----------+
| city | COUNT(*) | AVG(temp) | Var1 | Var_A | STD**2 | STD_A**2 |
+-----------+----------+-----------+------+-------+--------+----------+
| Paris | 2 | 20.0000 | 4.00 | 8.00 | 4.0000 | 8.0000 |
| Stockholm | 2 | 10.0000 | 4.00 | 8.00 | 4.0000 | 8.0000 |
| Paris | 2 | 20.0000 | 9.00 | 18.00 | 9.0000 | 18.0000 |
| Stockholm | 2 | 10.0000 | 9.00 | 18.00 | 9.0000 | 18.0000 |
| Paris | 2 | 20.0000 | 1.00 | 2.00 | 1.0000 | 2.0000 |
| Stockholm | 2 | 10.0000 | 1.00 | 2.00 | 1.0000 | 2.0000 |
+-----------+----------+-----------+------+-------+--------+----------+