**Statistics
**

**Ungrouped Data (Raw Data): **The information collected
systematically regarding a population or a sample survey is called an ungrouped
data. It is also called raw data.

**Grouped Data (Classified Data):** When a frequency distribution
is obtained by dividing an ungrouped data in a number of strata according to
the value of variate under study, such an information is called gruoped data or
classified data.

**Measures of central tendency:**

There are several measures of central tendency. Out of these, the following 3
are used more often

1. Mean, 2. Median and 3. Mode

**Ungroped Data**

**Mean of ungrouped data:**

Let x_{1}, x_{2}, x_{3}, ..., x_{n}
be n observations then mean is obtained by dividing the sum of n observations
by n. It is denoted by

X' = ^{åXi}/_{n}

For Eg: Find the mean of 4,6,8,6,7,8

Sol:

Mean X' = ^{åXi}/_{n}

= ^{(4 + 6 + 8 + 6 + 7 + 8)}/_{6}

= ^{39}/_{6}

= 6.5

**Median of ungrouped data:**

If the observations of an ungrouped data are arranged in increasing or
decreasing order of their magnitude, a value which divides these ordered
observations into two equal parts is called the median of the data. It is
denoted by M.

If the **number of observations** (**n**) is an **
odd** integer, then

M = Value of ^{(n+1)}/_{2} th observation in the arrangement of
observations in increaseing order.

If the **number of observations **(**n**) is
an **even** integer, then

M = ^{(Value of n/2 th observation + Value of (n/2
+ 1) th observation)}/_{2}

For Eg:

**Find the median of the following observations
4,6,8,6,7,8,8**

Sol: Observations in the ascending order are :

4, 6, 6, 7, 8, 8, 8

Here, n = 7 is odd.

Median :

M = Value of

= Value of

= Value of 4 th observation

= 7

**Mode of ungrouped data:
**An observation occurring most frequently in the data is called mode of
the data. It is denoted by Z.

For Eg:

4,6,8,6,7,8,8

In the given data, the observation 8 occurs maximum

number of times (3)

\ Mode(Z) = 8

1. The information regarding the no of children per family is given in the following table. Find the Mean, Median, Mode of the data

No of children | 0 | 1 | 2 | 3 | 4 | 5 |

No of families | 3 | 20 | 15 | 8 | 3 | 1 |

HINT: For input

For this type of example select X & Freq Option, Input the values like

X | Freq |

0 | 3 |

1 | 20 |

2 | 15 |

3 | 8 |

4 | 3 |

5 | 1 |

2. The frequency distribution of the marks obtained by 100 students in a test of Mathematics carrying 50 marks is given below. Find the Mean, Median, Mode of the data.

Marks obtained | 0 - 9 | 10 - 19 | 20 - 29 | 30 - 39 | 40 - 49 |

No of students | 8 | 15 | 20 | 45 | 12 |

HINT: For input

For this type of example select X,Y & Freq Option, Input the values like

X | Y | Freq |

0 | 9 | 8 |

10 | 19 | 15 |

20 | 29 | 20 |

30 | 39 | 45 |

40 | 49 | 12 |