What Is The First Quartile Of The Data Set Below 275 257

Hateful

Definition of Mean

The mean, also known every bit the "Arithmetic Mean" of a group of observations is the value that is equally shared out amid all the observations. In simple terms, it is the average of a set of numbers.

Examples of Mean

We use mean in referring to many values in our everyday lives.

For example,

Nosotros usually say that the average number of runs scored past a role player in this match was 20.

My average score in the examination was 65.

Symbol of Hateful

The symbol of mean is x̄.

Mean formula

The mean of a set of observations is calculated depending upon the grouping of numbers we have, i.e. whether the numbers whose means if to be determined are present in a grouped form or ungrouped form.

Mean of Ungrouped Data

Ungrouped information refers to the data where the numbers are present in a simple fashion, with each value divers separately. The arithmetic hateful of ungrouped data is defined as

Arithmetic Mean = $\frac{Sum\, of\, all\, observations}{Number\, of\, observations}$

Thus, if x₁, x₂, ten₃, ….…. x_north are the values of n observations then the arithmetic mean of these observations is given by

Arithmetics Mean = $\frac{x_1+ x_2+ x_3 + …+ x_n}{n}$

Example

If the heights of 5 persons are 144 cm, 152 cm, 151 cm, 158 cm and 155 cm respectively. Detect the mean height.

Solution

We accept been given that the heights of 5 persons are 144 cm, 152 cm, 151 cm, 158 cm and 155 cm respectively.

The observations, in this example, are – 144 cm, 152 cm, 151 cm, 158 cm and 155 cm

Full number of observations = five

To calculate the mean first nosotros need to observe the sum of all the observations.

Therefore,

(144 + 152 + 151 + 158 + 155) cm = 760 cm

Now, co-ordinate to the formula of the mean,

Arithmetic Mean = $\frac{Sum\, of\, all\, observations}{Number\, of\, observations}$

Therefore,

Arithmetic Mean = 760/5 cm = 152 cm

Hence, the hateful height of the 5 persons = 152 cm

Mean of Grouped Data

In the example of larger observations, data tin can be presented in the grade of a frequency table that exhibits the values taken by the variable and the respective frequencies. This grade of data is called grouped data or discrete frequency distribution.

If x₁, x₂, 10₃, ….…. x_{due north} are due north values of a variable with corresponding frequencies f₁, f₂, f_iii,………… f_northward respectively, then the arithmetic mean of these values is defined by

Mean = $\frac{f_1 x_1+ f_2 x_2+ f_3 x_3+ …+.f_n x_n}{f_1+ f_2+ f_3 + …+f_n}$

Mean = $\frac{\sum f_i x_i }{\sum f_i}$

where ∑ is called sigma which is a Greek alphabetic character that represents the summation.

The post-obit steps are involved in society to compute the arithmetic mean of grouped data –

Step1 Prepare a frequency table in such a style that its first column consists of the values of the variate and the 2d cavalcade the corresponding frequencies.

Step 2 Multiply the frequency of each row with the corresponding values of variate to obtain the third cavalcade containing f_ix_i

Step 3 Notice the sum of all entries in the third column to obtain $\sum f_i x_i$

Step 4 Observe the sum of all the frequencies in the second column to obtain $\sum f_i$or Due north

Step v Apply the formula: x̄ = $\frac{\sum f_i x_i }{\sum f_i}$

Case one

Given beneath are the ages of 25 students of a grade in a schoolhouse. Find the mean age of the students.

fifteen,16,16,14,17,17,16,15,fifteen,15,16,16,17,17,sixteen,xvi,17,17,16,16,sixteen,15,16,16,17

Solution

In lodge to find the mean, we must offset prepare a frequency table for the given data

The given data can be presented in the tabular form as:

In gild to compute the average age of students in the course, we prepare the post-obit table:

We have $\sum f_{i} = 25$ and $\sum f_{i}x_{i} = 400$

Therefore, Mean age =$\frac{\sum f_{i}x_{i} }{\sum f_{i}}=\frac{400}{25}= 16\,years$

Hence the mean of the 25 students is 16 years.

Case 2

The mean monthly wages of 10 members of a grouping is £1445. 1 more than fellow member whose monthly wage is £1500 has joined the group. Find the hateful monthly wage of 11 members.

Solution

Here the number of values, n = 10

The given hateful of these ten values x̄ = 1445

We know that

Arithmetic Hateful = $\frac{Sum\,of\,all\,observations}{Number\,of\,observations}$

This ways that 1445 = $\frac{\sum x_{i} }{ten}$

Hence,

$\sum x_{i}$ = 1445 x 10 = 14450

This ways that the total monthly wages of 10 persons = £14450

The monthly wage of one more person who has joined the grouping is given every bit £1500

Therefore,

The total monthly wages of 11 persons = £14450 + £1500 = £15950

And then, average monthly wages of 11 persons = $\frac{Sum\,of\,the\,monthly\,wages\,of\,11\,persons)}{Total\,number\,of\,persons}$

This means,

Average monthly wages of 11 persons = 15950/11 = £1450

Hence, the average monthly wage of 11 persons is £ 1450

Properties of Mean

The sum of the deviations of individual observations from the mean (x̄) is zero. Symbolically,

If 10₁, x₂, x₃, ………. x_n are the values of a variable and x̄ is their mean and then

$\sum_{i=0}^{n}(x_{i} – \bar{10})= 0$

If each observation in the data is replaced by mean (x̄) the full sum of all observations remain unaltered.
If every value of the variable 10 is increased (or decreased ) by a constant value, the arithmetic mean of the observation so obtained also increases (or decreases) by the same constant.
If the values of the variable x are multiplied ( or divided) by a constant the arithmetic mean of the new observations can be obtained by multiplying ( or dividing) the initial arithmetic mean by the same constant.

Definition

The median of a group of observations is the value of the variable which divides the group into two equal parts. In other words, the median is the value that exceeds and is exceeded by the aforementioned number of observations, i.e. it is the value such that the number of observations to a higher place it is equal to the number of observations below information technology.

In simple terms, the median is the value that lies in the middle of the data with one-half of the observations to a higher place it and the other half below it.

Median of Ungrouped Data

In the example of individual observations, i.e. ungrouped data, the following are used to find the median.

Stride ane: Suit the observations in ascending or descending order or magnitude.

Step ii: Determine the total number of observations, say, due north.

Footstep 3: If n is odd, then utilize the following formula:

Median = Value of $(\frac{due north+1}{ii})$^th observation

If n is even, then use the post-obit formula

Median = $\frac{Value\,of\,\frac{due north}{2}th\,observation\,+\,Value\,of\,(\frac{northward}{2}+i)th\,observation}{2}$

Permit us empathize both the scenarios with the help of examples

Example 1

Detect the median of the data: 24, 36, 46, 17, 18, 25, 35

Solution

We will detect the median using the above steps:

Step 1 Conform the observations in ascending or descending order or magnitude.

Therefore, let us conform the numbers in ascending order. We go:

17, xviii, 24, 25, 35, 36, 46

Step ii Decide the total number of observations, say, n.

Here the total number of observations northward is vii, which is an odd number

Footstep three If n is odd, then use the following formula:

Median = Value of $(\frac{n+1}{2})^{th}$ observation

Hence, we observe the value of $(\frac{northward+1}{2})^{th}$ observation = (\frac{seven+1}{ii})^{thursday} observation = 4^th observation

iv^th ascertainment in the list is 25

Hence the median of the given prepare of observations is 25.

Example 2

Discover the median of the data – 25, 34, 31, 23, 22, 26, 35, 28, xx, 32

Solution

We will observe the median using the following steps:

Step 1 Arrange the observations in ascending or descending order or magnitude.

Therefore, let united states of america arrange the numbers in ascending gild. We go:

twenty, 22, 23, 25, 26, 28, 31, 32, 34, 35

Step 2 Make up one's mind the total number of observations, say, north.

Here the total number of observations northward is 10, which is an even number

Footstep iii, If n is even, then use the post-obit formula

Median = $\frac{Value\,of\,\frac{n}{two}thursday\,observation\,+\,Value\,of\,(\frac{due north}{ii}+1)th\,observation}{2}$

At present, the value of $\frac{n}{two}thursday$ = value of $\frac{10}{ii}thursday$ ascertainment = value of v^th observation = 26

and

Value of $(\frac{n}{2}+1)$th observation = value of $(\frac{ten}{two}+1)$th ascertainment= value of vi^th observation = 28

Therefore, Median = $\frac{26+28}{ii}$ = $\frac{54}{ii}$ = 27

Hence, the median of the given observations = 27

Uses of Median

Median is the merely average that is used while dealing with qualitative data which cannot be measured quantitatively but still tin can be arranged in ascending or descending club of magnitude. For example, the median is used to find the boilerplate honesty, average intelligence, average beauty etc.
Median is used for determining typical value in bug concerning wages distribution of wealth etc.

Properties of Median

The median can be calculated graphically, while the mean cannot.
The sum of the accented deviations taken from the median is less than the sum of accented deviations from any other observations in the data.
The median is not much affected by farthermost values and would therefore exist a better representative every bit a eye or average.

Mode

Definition

Occasionally, when describing a set of information, the mode is used as a measure of key tendency. The fashion is the well-nigh frequently occurring value in a information gear up. In other words, the mode is of distribution is the value at the betoken around which the items tend to most heavily concentrated. Thus mode or the modal value is the value in a series of observations that occurs with the highest frequency.

For instance, allow the scores x students exist

45, 40, 44, 44, 43, 38, 44, 35, 18, 38

Permit us see which scores appears how many times in the List

Score	Number of times it appears
45	1
40	1
44	3
43	1
38	2
35	1
18	1

From the above tabular array, nosotros tin see that the mode is 44 which occurs thrice and the other scores occur only once or twice.

Way of Ungrouped Data

For ungrouped information, the style can be located just by inspecting the number of times each value appears in the set up. Hither the data tin exist arranged in an assortment and and so count the frequencies of each variate. The variate having maximum frequency is the mode.

The followings steps are used to find out the mode of ungrouped data:

Obtain the set up of observations.
Count the number of times the various values repeat themselves. In other words, gear up the frequency distribution table.
Find the value which occurs the maximum number of times. This means that observe the value which has the maximum frequency.
The value obtained in the previous stride is the fashion.

Instance

Discover the mode of the information given beneath:

4, 6, seven, 8, 12, 11, 13, 9, 13, nine, 13, 9, 9, seven, 8

Solution

On observing the information, we can place the number of times each value appears as its frequency in the table below:

Numbers	Frequency
4	1
6	1
7	2
8	2
ix	3
11	i
12	1
13	2

The above table clearly shows that the number nine occurs iii times which is the maximum amongst all frequencies.

Hence, the style for the given data set up is 9.

Uses of Mode

Mode is the average to be used to find the ideal size. For example, it is extensively used in business forecasting, in the manufacturing of dress, in the manufacturing of shoes etc.

Properties of Mode

Mode, like median, tin can be calculated graphically.
Mode is non affected by farthermost values.
The fashion tin can exist used for open-ended distribution and qualitative data.

Case one

The following number of goals were scored past a team in a series of x matches:

ii, three, 4, 5, 0, 1, iii, 3, 4, three

Find the mean, median and way of these scores.

Solution

Permit u.s.a. get-go calculate the mean of the given data.

Here, the observations in this case are – ii, 3, 4, 5, 0, 1, iii, 3, 4 and 3

Full number of observations = x

We know that to calculate the mean first we demand to discover the sum of all the observations.

Therefore,

(ii + iii + 4 + 5 + 0 +i + 3 + iii + iv +three) = 28

At present, according to the formula of the hateful,

Arithmetic Mean = $\frac{Sum\, of\, all\, observations}{Number\, of\, observations}$

Therefore,

Arithmetic Hateful = 28/x = 2.8

Hence, the hateful of the given data is 2.8

Now, permit us summate the median of the information

To calculate the median, we will first arrange the observations in ascending or descending order or magnitude.

We get,

0, 1, two, iii, iii, 3, 3, 4, 4, v

Now, we make up one's mind the total number of observations, say, n. Hither the total number of observations north is x, which is an even number.

If north is even, so we use the following formula

Median = $\frac{Value\,of\,\frac{n}{2}th\,observation\,+\,Value\,of\,(\frac{n}{2}+1)th\,observation}{2}$

Now, the value of $\frac{n}{2}$ thursday ascertainment = value of $\frac{x}{2}$ thursday observation = value of v^thursday observation = three

and

Value of $(\frac{n}{ii}+1)$ th observation = value of $(\frac{10}{2}+1)$ thursday observation = value of vi^th ascertainment = 3

Therefore, Median = $\frac{3+3}{ii}$ = $\frac{6}{2}$ = 3

Hence, median of the given observations = 3

Last, nosotros calculate the mode of the given data. Nosotros know that observation with the maximum frequency is called the mode.

On observing the data, we can place the number of times each value appears equally its frequency in the table beneath:

Number	Frequency
0	1
1	1
2	one
3	iv
four	2
v	1

The above table clearly shows that the number 3 occurs iv times which is the maximum amidst all frequencies.

Hence, the mode for the given data set is three

Mean, median and way are correlated by the following formula –

Style = 3 Median – 2 Mean

Example

If the ratio of mode and median of a certain data is half-dozen: 5, then find the ratio of its mean and median.

Solution

We know that the Empirical formula defining the relation between mean, median and mode is

Mode = three Median – 2 Hateful

We take been given that the ratio of mode and median of a certain data is 6: 5.

This ratio can also be written equally

$\frac{Mode}{Median} = \frac{half dozen}{five}$

Mode = $\frac{6\times Median}{five}$

Now, we supercede this value of mode in the empirical formula.

Nosotros become

$\frac{6\times Median}{v}$ = 3 Median – 2 Mean

⇒ 6 x Median = v ( 3 Median – 2 Mean)

⇒ 6 x Median = 15 Median – ten Mean

⇒ ten Mean = xv Median – vi Median

⇒ ten Mean = ix Median

Nosotros are required to find the ratio of mean and median. Hence nosotros will write the above equation every bit:

$\frac{Mean}{Median} = \frac{nine}{ten}$

Or Mean : Median = nine : ten

Hence, we can say that if the ratio of mode and median of a sure data is six: 5, then the ratio of its mean and median will be 9:10.

Recommended Worksheets

Solving Measures of Variability 6th Grade Math Worksheets
Measures of Variability (Basketball Themed) Worksheets
Solving Measures of Cardinal Tendency 6th Grade Math Worksheets

We spend a lot of time researching and compiling the information on this site. If you lot find this useful in your research, please utilize the tool below to properly link to or reference Helping with Math equally the source. We appreciate your support!