Does Pandas calculate ewm wrong?

There are several ways to initialize an exponential moving average, so I wouldn’t say pandas is doing it wrong, just different.

Here would be a way to calculate it like you want:

In [20]: s.head()
Out[20]: 
0    22.27
1    22.19
2    22.08
3    22.17
4    22.18
Name: Price, dtype: float64

In [21]: span = 10

In [22]: sma = s.rolling(window=span, min_periods=span).mean()[:span]

In [24]: rest = s[span:]

In [25]: pd.concat([sma, rest]).ewm(span=span, adjust=False).mean()
Out[25]: 
0           NaN
1           NaN
2           NaN
3           NaN
4           NaN
5           NaN
6           NaN
7           NaN
8           NaN
9     22.221000
10    22.208091
11    22.241165
12    22.266408
13    22.328879
14    22.516356
15    22.795200
16    22.968800
17    23.125382
18    23.275312
19    23.339801
20    23.427110
21    23.507635
22    23.533520
23    23.471062
24    23.403596
25    23.390215
26    23.261085
27    23.231797
28    23.080561
29    22.915004
Name: Price, dtype: float64