How to do linear regression, taking errorbars into account?

Not entirely sure if this is what you mean, but…using pandas, statsmodels, and patsy, we can compare an ordinary least-squares fit and a weighted least-squares fit which uses the inverse of the noise you provided as a weight matrix (statsmodels will complain about sample sizes < 20, by the way).

import pandas as pd
import numpy as np
import matplotlib.pyplot as plt
import matplotlib as mpl
mpl.rcParams['figure.dpi'] = 300

import statsmodels.formula.api as sm

x_list = [0.3333333333333333, 0.2886751345948129, 0.25, 0.23570226039551587, 0.22360679774997896, 0.20412414523193154, 0.2, 0.16666666666666666]
y_list = [0.13250359351851854, 0.12098339583333334, 0.12398501145833334, 0.09152715, 0.11167239583333334, 0.10876248333333333, 0.09814170444444444, 0.08560799305555555]
y_err = [0.003306749165349316, 0.003818446389148108, 0.0056036878203831785, 0.0036635292592592595, 0.0037034897788415424, 0.007576672222222223, 0.002981084130692832, 0.0034913019065973983]

# put x and y into a pandas DataFrame, and the weights into a Series
ws = pd.DataFrame({
    'x': x_list,
    'y': y_list
})
weights = pd.Series(y_err)

wls_fit = sm.wls('x ~ y', data=ws, weights=1 / weights).fit()
ols_fit = sm.ols('x ~ y', data=ws).fit()

# show the fit summary by calling wls_fit.summary()
# wls fit r-squared is 0.754
# ols fit r-squared is 0.701

# let's plot our data
plt.clf()
fig = plt.figure()
ax = fig.add_subplot(111, facecolor='w')
ws.plot(
    kind='scatter',
    x='x',
    y='y',
    style='o',
    alpha=1.,
    ax=ax,
    title='x vs y scatter',
    edgecolor='#ff8300',
    s=40
)

# weighted prediction
wp, = ax.plot(
    wls_fit.predict(),
    ws['y'],
    color='#e55ea2',
    lw=1.,
    alpha=1.0,
)
# unweighted prediction
op, = ax.plot(  
    ols_fit.predict(),
    ws['y'],
    color='k',
    ls='solid',
    lw=1,
    alpha=1.0,
)
leg = plt.legend(
    (op, wp),
    ('Ordinary Least Squares', 'Weighted Least Squares'),
    loc='upper left',
    fontsize=8)

plt.tight_layout()
fig.set_size_inches(6.40, 5.12)
plt.show()

WLS residuals:

[0.025624005084707302,
 0.013611438189866154,
 -0.033569595462217161,
 0.044110895217014695,
 -0.025071632845910546,
 -0.036308252199571928,
 -0.010335514810672464,
 -0.0081511479431851663]

The mean squared error of the residuals for the weighted fit (wls_fit.mse_resid or wls_fit.scale) is 0.22964802498892287, and the r-squared value of the fit is 0.754.

You can obtain a wealth of data about the fits by calling their summary() method, and/or doing dir(wls_fit), if you need a list of every available property and method.

I wrote a concise function to perform the weighted linear regression of a data set, which is a direct translation of GSL's "gsl_fit_wlinear" function. This is useful if you want to know exactly what your function is doing when it performs the fit

def wlinear_fit (x,y,w) :
    """
    Fit (x,y,w) to a linear function, using exact formulae for weighted linear
    regression. This code was translated from the GNU Scientific Library (GSL),
    it is an exact copy of the function gsl_fit_wlinear.
    """
    # compute the weighted means and weighted deviations from the means
    # wm denotes a "weighted mean", wm(f) = (sum_i w_i f_i) / (sum_i w_i)
    W = np.sum(w)
    wm_x = np.average(x,weights=w)
    wm_y = np.average(y,weights=w)
    dx = x-wm_x
    dy = y-wm_y
    wm_dx2 = np.average(dx**2,weights=w)
    wm_dxdy = np.average(dx*dy,weights=w)
    # In terms of y = a + b x
    b = wm_dxdy / wm_dx2
    a = wm_y - wm_x*b
    cov_00 = (1.0/W) * (1.0 + wm_x**2/wm_dx2)
    cov_11 = 1.0 / (W*wm_dx2)
    cov_01 = -wm_x / (W*wm_dx2)
    # Compute chi^2 = \sum w_i (y_i - (a + b * x_i))^2
    chi2 = np.sum (w * (y-(a+b*x))**2)
    return a,b,cov_00,cov_11,cov_01,chi2

To perform your fit, you would do

a,b,cov_00,cov_11,cov_01,chi2 = wlinear_fit(x_list,y_list,1.0/y_err**2)

Which will return the best estimate for the coefficients a (the intercept) and b (the slope) of the linear regression, along with the elements of the covariance matrix cov_00, cov_01 and cov_11. The best estimate on the error on a is then the square root of cov_00 and the one on b is the square root of cov_11. The weighted sum of the residuals is returned in the chi2 variable.

IMPORTANT: this function accepts inverse variances, not the inverse standard deviations as the weights for the data points.