# The following is a frequency table of the score obtained in a science quiz competition. Find the median score.

Score $10$ $15$ $20$ $25$ $30$ Frequency $2$ $3$ $5$ $6$ $4$

${\text{A) 22}}$

${\text{B) 22}}{\text{.5}}$

${\text{C) 20}}$

${\text{D) 25}}$

Score | $10$ | $15$ | $20$ | $25$ | $30$ |

Frequency | $2$ | $3$ | $5$ | $6$ | $4$ |

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**Hint:**First we have to find the sum of scores \[\left( {\text{x}} \right)\] and the frequency \[\left( {\text{f}} \right)\]as this a discrete frequency distribution. Then by using the formula, whichever is around the answer i.e., round off is the required answer.

**Formula used:**When the number of observation is odd:

Median = ${\left( {\dfrac{{{\text{N + 1}}}}{2}} \right)^{th}}$ term

When the number of observation is even:

First, find ${\left( {\dfrac{{\text{N}}}{2}} \right)^{th}}$ term

Then ${\left( {\dfrac{{{\text{N + 1}}}}{2}} \right)^{th}}$ term

And find the average of two values i.e., average of ${\left( {\dfrac{{\text{N}}}{2}} \right)^{th}}$term and ${\left( {\dfrac{{{\text{N + 1}}}}{2}} \right)^{th}}$ term

**Complete step-by-step solution:**

First we have to rearrange the given data as follows:

Score \[\left( {\text{x}} \right)\] | Frequency \[\left( {\text{f}} \right)\] |

$10$ | $2$ |

$15$ | $3$ |

$20$ | $5$ |

$25$ | $6$ |

$30$ | $4$ |

${\text{N = 20}}$ |

Since the number of observation is even, we need to find average of ${\left( {\dfrac{{\text{N}}}{2}} \right)^{th}}$ term and ${\left( {\dfrac{{{\text{N + 1}}}}{2}} \right)^{th}}$ term

On putting the value of N and we get,

${\left( {\dfrac{{\text{N}}}{2}} \right)^{th}} = \dfrac{{20}}{2}$

Let us divide the term and we get

$ \Rightarrow {10^{th}}$ observation

Now we have to find:

${\left( {\dfrac{{{\text{N + 1}}}}{2}} \right)^{th}} = \dfrac{{20 + 1}}{2}$

On dividing the term and we get,

$ \Rightarrow 10.5$

Taking as round value and we get,

$ \Rightarrow {11^{th}}$ Observation

Now the changed data as formed as follows:

Score \[\left( {\text{x}} \right)\] | Frequency \[\left( {\text{f}} \right)\] | \[\left( {{\text{cf}}} \right)\] |

$10$ | $2$ | $2$ |

$15$ | $3$ | $5$ |

$20$ | $5$ | $10$ |

$25$ | $6$ | $16$ |

$30$ | $4$ | $20$ |

${\text{N = 20}}$ |

So here, ${10^{th}}$ term lies in $20$ and ${11^{th}}$ term lies in $25$

So we can write it as, by using the formula and find the median

Median = $\dfrac{{20 + 25}}{2}$

Let us add the numerator and we get,

Median = $\dfrac{{45}}{2}$

Let us divide the term and we get,

Median =$22.5$

**Therefore the correct answer is ${\text{B) 22}}{\text{.5}}$.**

**Note:**In this question we have an alternative method.

Alternative method:

We can also find median in a simple way.

Score \[\left( {\text{x}} \right)\] | Frequency \[\left( {\text{f}} \right)\] |

$10$ | $2$ |

$15$ | $3$ |

$20$ | $5$ |

$25$ | $6$ |

$30$ | $4$ |

Here we can also write this elaborately since it is discrete distribution,$10,10,15,15,15,20,20,20,20,20,25,25,25,25,25,25,30,30,30,30$

Median is the middle value of the given observation, so in this observation there are two numbers in middle (since the total numbers are even)

They are $20$ and $25$

Median = average of these two numbers

Median = $\dfrac{{20 + 25}}{2}$

Let us add the numerator and we get,

Median = $\dfrac{{45}}{2}$

Let us divide the term and we get,

Median =$22.5$

Therefore the correct answer is ${\text{B) 22}}{\text{.5}}$.