I don't think it can be done, if if you would know the weights. If you know the weights, you can still not reverse engineer anything about the underlying distribution. However, you can back out parameters from the maximum and minimum price because we know the sample size. It seems your sample size is 37. So you can treat the minimum and maximum prices as (very rough) estimates of quantiles. Combining the quantiles with an assumption about the type of distribution, should allow you to back out the parameters of this distribution.

ISSUE: How to determine weighted variance from a weight mean with weight information not given?

BASIS FOR WEIGHT: There are three bases for weight from the information given. These bases are: (i) number of deals, (ii) number of volume, and (iii) number of turnover. The respective weight of these three items follows:

(a) DEALS: 37 + 35 = 72. The weights are 37/72 = 0.51 and 35/72 = 0.49.

(b) VOLUME: 41,0000 + 5,100 = 92,100. The weights are 41,000/92,100 = 0.45 and 5,100/92,100 = 0.55.

(c) TURNOVER: 1,923,275 + 2,402,550 = 4,325,825. The weights are 0.80 and 0.20.

Let’s called the two strings of data as 1X and 2X. The mean for 1X is given as 46.94 and the mean for 2X is given as 47.04. Now, the mean of these two items is (46.94 + 47.04)/2 = 47.49.

DETERMINE WEIGHTED MEAN: Let us use the number of deals as the illustration (volume and turnover weight can also be repeated with the same process). The weight mean may be determine in the following steps:

Step 1: The weighted mean is given by:

(1)   X^w = sum(wi)(X^i)

Recall that 1X = 46.94; w1 = 0.51 and 2X = 47.04 and w1 = 0.49. The calculation follows:

0.51(46.94) + 0.49(47.04)

=23.94 + 23.27

=47.21

The weighted mean is now X^w = 47.21.

Step 2: Since the original sample size was 72. Arguendo assume that the data is normally distribute. Thus, the expected value of X or E(Xi) = mu, and the expected weighted mean is equal to the sum of weights theoretical mean or:

(2)   E(X^w) = sum(wi)(mui)

Since mu^ = mu, the expected E(X^w) = mu. The objective is to determine mu. Recall that:

(3)   T = (X^ - mu) / (S/sqrt(n))

Therefore, mu is given by:

(4)   Mu = X^ - t[S(sqrt(n))]

Where n = 2 coming from two average price given at the beginning, and X^ = 47.20. Arguendo assume t(df = infinity) = 1.64 We can determine the value of mu as:

Mu = X^ - t[S(sqrt(n))]

=47.21 – 1.64(0.2256/sqrt(2))

=47.21 – 1.64(0.2256/1.41)

=47.21 – 1.64(0.16)

=47.21 – 0.2624

=46.94

Note that S = 0.2256 came from OLS of these two items: 46.94 and 47.04 compared against the mean of 47.21.

Step 3: determine the weight sample mean. Recall that E(X^w) = sum(wi)(mui); therefore:

Sum(X^) = 0.51(46.94) + 0.49(46.94)

=23.94 + 24.47

=48.41

The weighted theoretical mean is 48.41.

WEIGHTED VARIANCE: The weighted variance is given by:

(5)   Sig2X^w = sum(w2)(sig2)

Step 4: Recall that the Z-equation for the theoretical unit normal distribution is given as:

(6)   Z = (X^ - mu) (sig / sqrt(n))

Therefore, sigma is written as:

(7)   Sig = [(X^ - mu) / Z]sqrt(n)

Since we used t = 1.64; for Z, we will fix it at Z = 1.65, i.e. 95% confidence interval. The calculation for sigma follows:

Sig = [(X^ - mu) / Z]sqrt(n)

=[(47.21 – 46.94) / 1.65]sqrt(2)

=(0.27/1.65)1.41

=0.1636/1.41

=0.1161

Sigma is the theoretical standard deviation. Recall that in OLS, the standard deviation is the square root of the variance. Therefore, variance is the square of sigma.

The weighted variance is given as:

Sig2X^w = sum(w2)(sig2)

=0.51(0.0135)+0.49(0.0135)

=0.0069 + 0.00662

=0.01351

THEREFORE, the weighted variance is 0.0135 and the weighted standard deviation is 0.1161. The determination of variance and standard deviation can also be made on the basis of volume and turnover in the same steps outline above. These three sources of weighted variance and standard deviation can then be compared.

If you are interested in the computation of the TIME variance, which is typically the case in models like CAPM, then you can take the weighted price or the mid price (HIGH-LOW)/2 as a "daily price" and compute the variance in an appropriate time wondow. This is tipically referred as "historical volatility". If you are interested in the comptation of the intra-day variance, that is the variance of the prices of the different transactions inside the same trading session, then you can't do it with your data.  BUT you can obtain a reliable estimate of the "dispersion" of intra-day prices simply taking the difference HIGH-LOW. This measure is probably more robust than the variance itself.