**1. The scores in mathematics test (out of 25) of 15 students is as follows:**

**19, 25, 23, 20, 9, 20, 15, 10, 5, 16, 25, 20, 24, 12, 20**

**Find the mode and median of this data. Are they same?**

**Solution:**

Firstly, we arrange the given scores in ascending order,

We get

5, 9, 10, 12, 15, 16, 19, 20, 20, 20, 20, 23, 24, 25, 25

**Mode**,

According to the definition of Mode, We know that Mode is the value of the variable which occurs most frequently.

Clearly, 20 occurs the maximum number of times.

Hence, the mode of the given scores is 20

Median,

According to the definition of Median, we know that the value of the middle-most observation is called the median of the data.

Here n = 15, which is odd.

Where n is total the number of students.

Now,

median = value of ½ (n + 1)^{th} observation.

= ½ (15 + 1)

= ½ (16)

= 16/2

= 8

Then, value of 8^{th} term = 20

Hence, the median is 20.

Yes, both the values are same.

**2. The runs scored in a cricket match by 11 players is as follows:**

**6, 15, 120, 50, 100, 80, 10, 15, 8, 10, 15**

**Find the mean, mode and median of this data. Are the three same?**

**Solution:**

Firstly, we arrange the runs scored in ascending order.

we get,

6, 8, 10, 10, 15, 15, 15, 50, 80, 100, 120

Mean,

Mean of the given data = Sum of all observations/ Total number of observations

= (6 + 8 + 10 + 10 + 15 + 15 + 15 + 50 + 80 + 100 + 120)/ 11

= 429/11

= 39

Mode,

According to the definition of Mode, We know that Mode is the value of the variable which occurs most frequently.

Clearly, 15 occurs the maximum number of times.

Hence, the mode of the given sores is 15

Median,

We know that the value of the middle-most observation is called the median of the data.

Here n = 11, which is odd.

Where n is the total number of players.

Now,

median = value of ½ (n + 1)^{th} observation.

= ½ (11 + 1)

= ½ (12)

= 12/2

= 6

Then, value of 6^{th} term = 15

Hence, the median is 15.

No, these three are not same.

**3. The weights (in kg.) of 15 students of a class are:**

**38, 42, 35, 37, 45, 50, 32, 43, 43, 40, 36, 38, 43, 38, 47**

**(i) Find the mode and median of this data.**

**(ii) Is there more than one mode?**

**Solution:**

Firstly, we arrange the given weights of 15 students of a class in ascending order.

we get,

32, 35, 36, 37, 38, 38, 38, 40, 42, 43, 43, 43, 45, 47, 50

**(i) Mode and Median**

**Mode,**

According to the definition of Mode, We know that Mode is the value of the variable which occurs most frequently.

Clearly, 38 and 43 both occurs 3 times.

Hence, modes of the given weights are 38 and 43.

**Median**,

We know that the value of the middle-most observation is called the median of the data.

Here n = 15, which is odd.

Where n is the number of students.

Now,

median = value of ½ (n + 1)^{th} observation.

= ½ (15 + 1)

= ½ (16)

= 16/2

= 8

Then, value of 8^{th} term = 40

Hence, the median is 40.

**(ii) Is there more than one mode? **

Yes, there are 2 modes for the given weights of the students.

**4. Find the mode and median of the data: 13, 16, 12, 14, 19, 12, 14, 13, 14**

**Solution:**

Firstly, we arrange the given data in ascending order.

we get,

= 12, 12, 13, 13, 14, 14, 14, 16, 19

**Mode**,

Clearly, 14 occurs the maximum number of times.

Hence, the mode of the given data is 14.

**Median**,

We know that the value of the middle-most observation is called the median of the data.

Here n = 9, which is odd.

Where n is the number of students.

Now,

median = value of ½ (9 + 1)^{th} observation.

= ½ (9 + 1)

= ½ (10)

= 10/2

= 5

Then, value of 5^{th} term = 14

Hence, the median is 14.

**5. Tell whether the statement is true or false:**

**(i) The mode is always one of the numbers in a data.**

**Solution:**

The given statement is true.

It is because Mode is the value of the variable which occurs most frequently in the given data.

**(ii) The mean is one of the numbers in a data.**

**Solution:**

The given statement is false.

It is because mean is may be or may not be one of the numbers in a data.

**(iii) The median is always one of the numbers in a data.**

**Solution:**

The given statement is true.

It is because the median is the value of the middle-most observation in the given data while arranged in ascending or descending order.

**(iv) The data 6, 4, 3, 8, 9, 12, 13, 9 has mean 9.**

**Solution:**

Mean = Sum of all given observations/ number of observations

= (6 + 4 + 3 + 8 + 9 + 12 + 13 + 9)/8

= (64/8)

= 8

Hence, the given statement is false.

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