How can I create a continuous distribution of a dataset?

Question

I wish to create a continuous probability distribution from this dataset.

The 'Value' shows a measured value and the 'Weight' is the probability of measuring this value in this measurement.

I already graphed the data. On the x-axis it shows the value, and the Y-axis the probability. But I wish to create an exact distribution to fit this data.

In my data-analysis I eventually wish to compare several data distributions by their parameters. I hope you guys can help me out.

Line #	Value	Weight
0	0.0538502	0.016508
1	0.0184823	0.0298487
2	0.0647929	0.0122637
3	0.0262852	0.0234716
4	0.0447611	0.0197072
5	0.0643164	0.0165399
6	0.0709176	0.0143751
7	0.0871276	0.012253
8	0.0341064	0.0197392
9	0.0593696	0.0143858
10	0.0436119	0.0202617
11	0.0505131	0.0191846
12	0.0378706	0.0207842
13	0.0298233	0.0250712
14	0.157727	0.0111866
15	0.0556603	0.0186408
16	0.0542849	0.017617
17	0.0395772	0.0180969
18	0.0694962	0.0117305
19	0.0343318	0.0229277
20	0.139291	0.00907511
22	0.0232517	0.0186514
23	0.207768	0.0069423
24	0.0156452	0.021872
25	0.117749	0.0100989
26	0.124017	0.0111973
27	0.0679313	0.0133407
28	0.0733413	0.0117198
29	0.100553	0.0133407
30	0.0695865	0.016508
31	0.117732	0.0138633
32	0.0540577	0.0170518
33	0.0736274	0.0170625
34	0.0332381	0.0293155
35	0.0803423	0.0159961
36	0.0465	0.0191846
37	0.0889299	0.0159854
38	0.053232	0.020251
39	0.131361	0.0122637
40	0.0233194	0.0240048
41	0.830735	0.0053107
42	0.341012	0.0069423
43	0.101263	0.0106534
44	0.127061	0.00959765
45	0.13706	0.0122637
46	0.120035	0.0106641
47	0.0801194	0.0138526
48	0.0617996	0.0165186
49	0.197555	0.0117305
50	0.0810635	0.0133301
51	0.0178539	0.0335811
52	0.0391433	0.0170518
53	0.0663863	0.0133194
54	0.0617675	0.0170625
55	0.00684359	0.0346582
56	0.0642299	0.0133301
57	0.00970105	0.0239941
58	0.0307687	0.0213068
59	0.0160796	0.0255937
60	0.0147901	0.0266388
61	0.073745	0.0122637
62	0.0420728	0.0207949
63	0.0211625	0.0207949
66	0.0241562	0.0255937
67	0.0329688	0.0239834
68	0.0739628	0.0181289
69	0.0149927	0.0266388
70	0.0130271	0.0378467
73	0.0107957	0.0351914
74	0.040447	0.0175744
75	0.00123215	0.0559756
76	0.0134575	0.0309151
77	0.00592594	0.0453116

Answer 1

It looks like the data you have is a sort of (non-normalized) histogram.

The first task is of course to plot it:

df = df.sort_values('Value')
plt.plot(df['Value'], df['Weight'])
plt.xlabel('value')
plt.ylabel('weight')

At first glance, it could indicate an exponential or a power-law distribution, but let's see.

Let's first try to smooth out that curve:

import statsmodels.api as sm

x, w = df['Value'].values, df['Weight'].values
s = pd.DataFrame(sm.nonparametric.lowess(w, x, frac=0.2), columns=['x', 'w']).set_index('x').squeeze()
s = s.reindex(np.linspace(x.min(), x.max(), 200), method='ffill', limit=1).interpolate()
s.plot()
plt.plot(x, w, '.')

That gives an okay-ish fit:

We can then use that to generate a fake, crude "sample" following that smooth pdf:

sample = np.random.choice(s.index, p=s/s.sum(), size=1000)

At that point, you can make QQ plots with various distributions following your intuition, and select one that seems to fit well:

from scipy.stats import _continuous_distns as distns

# trying a normal (the default)
sm.qqplot(sample, line='q')
plt.title('Normal')

Clearly not a good fit at all (but we knew that from a first glance at the data):

# trying an exponential
sm.qqplot(sample, distns.expon, line='q')
plt.title('Exponential')

Not very good either:

Perhaps a power-law would fit better?

# we are only interested in the parameter a, so we are
# not going to let loc and scale be fitted;
# instead, we will freeze them at loc=0, scale=1
a, loc, size = distns.powerlaw.fit(sample, floc=0, fscale=1)

# then, we do the QQ plot with the fitted parameter a
sm.qqplot(sample, distns.powerlaw, distargs=(a,), line='q')
plt.title(f'Power law with a={a}')

Corresponding distribution and how to use it

You can now instantiate a distribution following what was found (type and parameters), draw random variates from it, and also plot the pdf directly for comparison purposes with the original data:

g = distns.powerlaw(a=a)

# new points drawn according to g
v = g.rvs(size=100000)
plt.hist(v, bins=100, density=True, histtype='step');

Direct pdf plot and comparison with the original data:

y = g.pdf(x)
plt.plot(x, y/y.sum())
plt.plot(x, w/w.sum(), '.')
plt.title('Normalized pdf and original sample data')

Last word

So, where to go from here? You should look in depth into that distribution and its physical meaning, and see if that makes sense in the context of your experimental setup.