# Why do we find central tendencies tendencies for grouped frequency data?

When data is grouped, it does not give the exact values of the frequencies of the observations. Then how do we calculate the Central Tendencies of the data? Our school text books give us some formulas. For arithmetic mean, we take the class mark(midpoint) of the class as the value of all observations, and then multiply all class marks with the frequencies and divide the sum by the sum of frequencies. But here, it is not necessary that the values are anywhere near the midpoint. They could be absolutely anywhere and in a data where class size is large, the mean will be very different from what it is supposed to be. For median and mode we have other formulas too. Mode troubles me very much. $$\text{Mode} = L \ + \frac{f_{m}-f_{m-1}}{(f_{m}-f_{m-1}) + f_{m}-f_{m+1}} \cdot w$$ where,
• L is the lower class boundary of the modal group.
• fm-1 is the frequency of the group before the modal group.
• fm is the frequency of the modal group.
• fm+1 is the frequency of the group after the modal group.
• w is the group width. Why do we even have a mode formula if we get an answer with decimals in a data of family size of families in a locality?(Such a value does not actually exist, hence it will be having frequency zero which in no way is the maximum) I find it absurd to have any formula for these central tendencies in grouped data. It is bound to be wrong(if someone else has the actual discrete data and can judge the answer) and hence will be inherently useless. So how and why the formula? What can I do with the mode so obtained when it is not fulfilling the only thing it is supposed to do? Please don't give monstrous derivations I'm only 10 grade.