Simulate incomes only from median

I am looking to generate data for income by using lognormal distribution but unfortunately, I have only median of incomes. Do you have any idea how can I generate data for incomes or I should have also mean and std?

The Log Normal Distribution requires two parameters, $$\mu$$ and $$\sigma^2$$, be specified.

We note that:

The median is defined as $$\text{exp}(\mu)$$

Given that you have an estimate for the median - you can determine $$\mu$$ by taking the log of the median. However, you will then need to specify a value $$\sigma^2$$ in order to generate a random sample $$X_1, \dots, X_n$$ where $$X_i \sim Lognormal(\mu, \sigma^2)$$.

You could then use R or Python to generate some sample data. Here is an example of how to generate a sample using R.

# Specify mu and sigma
mu = 1
sd = 0.25

# Generate Sample
sample = rlnorm(10000, meanlog = mu, sdlog = sd)

# Calculate the estimated mean and median and compare against the five point summary.
mean = exp(mu + ((sd)^2)/2 )
median = exp(mu)

print(mean)
print(median)
summary(sample)