Solutions

Calculate $\sum x_i y_i$

In [6]:
sum_x_y = np.sum((bolt_df.distance * bolt_df.time))
print(sum_x_y)

3829.69710839

Calculate $m$ and $c$

From the equations for $m$ and $c$ above

In [7]:
N = len(bolt_df.time)
print(N)

11
In [8]:
N = len(bolt_df.time) ## we also need the number of points
m = ((N * sum_x_y) - (sum_x*sum_y))/((N * sum_x_2) - (sum_x)**2)
c = (sum_y - m*sum_x)/N
print(m,c)

0.10015849092636359 -0.04800871322726992
In [9]:
fig = plt.figure(figsize=(10,6))
ax = fig.add_subplot(1,1,1)
ax.errorbar(bolt_df.distance, bolt_df.time, yerr=bolt_df.err_time, color='k', marker='o', linestyle='None', label='Data')
ax.set_xlabel('Distance (m)')
ax.plot(bolt_df.distance, m*bolt_df.distance + c, color='blue', ls='--', label='$y = m x + c$')
ax.set_ylabel('Time (s)')
plt.legend()
title_string = "$m = {0:.3f}$, $c = {1:.3f}$".format(m, c)
ax.set_title(title_string)
plt.show()

Check your result

In [10]:
def line(x,m,c):
    return m*x + c
In [11]:
popt, pcov = curve_fit(line, bolt_df.distance, bolt_df.time)
m_fit = popt[0]
c_fit = popt[1]
In [12]:
print("My results: m = {0:.5f}, c = {1:.5f}".format(m, c))
print("Curve fit: m = {0:.5f}, c = {1:.5f}".format(m_fit, c_fit))

My results: m = 0.10016, c = -0.04801
Curve fit: m = 0.10016, c = -0.04801

Exactly the same!