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I have run a Kaplan Meier estimator with the survival package and it gives a median.

survfit(Surv(time, status) ~ 1, data = df)

Call: survfit(formula = Surv(time, status) ~ 1, data = data)

    n  events  median 0.95LCL 0.95UCL 
   125     33    76     51      88

Is this median the median survival rate? And if so what does it tell me? Does it give the median time individuals survive? Or is it the time when the survival probability for individuals is at 0.5? Could somebody please clarify?

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The time at which the Kaplan-Meier survival curve crosses the 50% line is the non-parametric estimate of the median survival time. Therefore if the curve never crosses 0.5 a median survival time cannot be calculated. Median survival time + 95% confidence interval are computed according to Brookmeyer & Crowley, 1982. So the 5th question of yours gives the best interpretation:

Or is it the time when the survival probability for individuals is at 0.5?

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