A dataset consists of five observations: $19, 15, 24, 26$ and $x$. If the median of the dataset is the same as the median of the three numbers: $24, 26,$ and $x,$ then the mean of the dataset is

The dataset consists of five observations: $19, 15, 24, 26$ and $x$.

The median of all five numbers must equal the median of $(24, 26, x)$.

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For three numbers $(24, 26, x)$, the median depends on where $x$ falls:

• If $x \leq 24$: median is $24$

• If $24 < x < 26$: median is $x$

• If $x \geq 26$: median is $26$

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The known values in order are: $15, 19, 24, 26$

For five numbers, the median is the third value when arranged in order.

Case 1: If $x \leq 24$, the median of $(24, 26, x)$ is $24$

When $x = 24$: The five numbers become $(15, 19, 24, 24, 26)$

Median $= 24$ ✓

Case 2: If $24 < x < 26$, the median of $(24, 26, x)$ is $x$

The five numbers: $(15, 19, 24, x, 26)$

Median $= 24 \neq x$ ✗

Case 3: If $x \geq 26$, the median of $(24, 26, x)$ is $26$

The five numbers: $(15, 19, 24, 26, x)$

Median $= 24 \neq 26$ ✗

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Therefore $x = 24$.

The dataset is: $19, 15, 24, 26, 24$

Mean $= \dfrac{19 + 15 + 24 + 26 + 24}{5}$

$= \dfrac{108}{5}$

$= 21.6$