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# Histograms: Estimating the Median

### Histograms: Estimating the Median

We can use a special method called linear interpolation to find the median from a histogram.

Histograms are plotted from grouped data, so we can't be sure of the individual data they represent. We use a special method to **estimate** the mean since we cannot observe it directly.

This method is called **linear interpolation**. It involves finding a proportional relationship between the **position** and **value** of the median within its grouped interval.

What is the correct formula to find the frequency after observing a histogram?

Here is a table with the intervals, frequency densities and frequencies

We can calculate the frequency with the formula $frequency \space density \times class \space width$.

What is the frequency at the interval $155\leq m <175$?

The frequency is 18 at the interval $155\leq m <175$

$1.2 \times 15 = 18$

The cumulative frequency can identify the median

We need to identify **which interval** the median value is located in.

What is the missing cumulative frequency value?

The median is the 23rd value

Therefore, we can see that it is located in the interval $155 \leq m <170$. The 23rd student, therefore, has a height between 155cm and 170cm.

We can represent this on a number line

The number line presents a schematic showing where the median lies relative to the upper and lower bounds of the interval.

Now we can identify a proportional relationship

The proportion of the **position** of the median is the same as the proportion of the **value** of the median (relative to the upper and lower bounds of the interval): $\dfrac{23-14}{32-14}=\dfrac{Q2-155}{170-155}$

We can simplify this equation

\dfrac{23-14}{32-14}=\dfrac{Q2-155}{170-155} \space \rightarrow \space $$\frac{9}{18}=\frac{Q2-155}{15}

By rearranging, we can find the median

$\dfrac{9}{18}=\dfrac{Q2-155}{15} \space \rightarrow \space 7.5=Q2-155$

If $7.5=Q2-155$, what is $Q2$?

The median is $162.5cm$

Great work! Remember this is only an estimate, since we cannot observe the median directly in grouped intervals.