import pandas as pd
from pandas_datareader import data as pdr
import numpy as np
import datetime

end = datetime.date.today()
begin=end-pd.DateOffset(365*10)
st=begin.strftime('%Y-%m-%d')
ed=end.strftime('%Y-%m-%d')


data = pdr.get_data_yahoo("AAPL",st,ed)

def bollinger_strat(data, window, no_of_std):
    rolling_mean = data['Close'].rolling(window).mean()
    rolling_std = data['Close'].rolling(window).std()

    df['Bollinger High'] = rolling_mean + (rolling_std * no_of_std)
    df['Bollinger Low'] = rolling_mean - (rolling_std * no_of_std)     

bollinger_strat(data,20,2)

This is probably due to Pandas normalizing by N - 1 instead of N. See the first note at https://pandas.pydata.org/docs/reference/api/pandas.core.window.rolling.Rolling.std.html

You can change this behavior using the degrees of freedom argument, ddof, e.g. pd.Series(a).rolling(5).std(ddof=0).

It seems that implementation of std() in pd.rolling prefers high performance over numerical accuracy. However You can apply np version of standard deviation:

df.loc[:, 'std'] = df.rolling(window, min_periods=1).apply(np.std)

Result:

    values          std
0   1234.0     0.000000
1   4567.0  1666.500000
2   6800.0  2287.053757
3   6810.0  2280.329732
4   6821.0  2186.267193
5   6820.0   898.332366
6   6820.0     8.158431
7   6820.0     4.118252
8   6820.0     0.400000
9   6820.0     0.000000
10  6820.0     0.000000

Now precision is better.