## Pre-Requisires

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**Statistics**

# Statistics is the study of collection, organization, analysis and interpretation of data.

**Data**

# Data is a distinct piece of information in the form of fact or figures collected or represented for any specific purpose. The word data is derived from the Latin word **Datum**.

**Collection of Data**

# In general, data is of two types. They are:

- Primary Data
- Secondary Data

**Primary Data**

# Primary data is the data collected from any firsthand experience for an explicit use or purpose.

**Secondary data**

# Secondary data is the data collected by any third party for a different purpose other than the user.

**Presentation of Data**

# After collecting data it is important to present it meaningfully. There are many ways to present data.

**1. Raw Data or Ungrouped Data**

# a. Raw Data or Ungrouped Data is the collected data without any change in its form.

**Example**

# The marks obtained by 10 students in a Mathematics test are:

# 55 **36 95** 73 60 42 25 78 75 62

**Range**

# Range is the difference between the highest and the lowest values of data.

For Above data: **Range = 95 – 36 = 59**

# b. **Frequency Distribution Table **– Frequency Distribution Table is the data of a large number of items converted into tabular form.

**Frequency is the number of times the item comes to the table.**

**2. Grouped Data**

# To present the very large number of items in the data we use a grouped distribution table.

**Grouped Distribution Table**

# a. **Class Interval** ā The group used to classify the data is called the **class interval** i.e. 20 ā 30, 30 ā 40.

# b. **Upper Limit** – In each class interval, the greatest number is the **upper-class limit**.

# c. **Lower Limit** ā In each class interval, the smallest number is the lower class limit.

# d. **Class Size** – It is the difference between the upper limit and the lower limit i.e. 10.

**Class Size = **Upper Limit – Lower Limit

# e. **Class Mark ** ā The midpoint of each class interval is the class mark.

**Grouped data could be of two types as below:-**

- Inclusive or discontinuous Frequency Distribution.
- Exclusive or continuous Frequency Distribution

**Inclusive or discontinuous Frequency Distribution** ā If the upper limit of a class is different from the lower limit of its next class then it is said to be an Inclusive or discontinuous Frequency Distribution.

**Example**

# Draw the histogram of the following frequency distribution.

Daily earnings (in Rs) | 700 ā 749 | 750 ā 799 | 800 ā 849 | 850 ā 899 | 900 ā 949 | 950 ā 999 |

No. of stores | 6 | 9 | 2 | 7 | 11 | 5 |

**Exclusive or continuous Frequency Distribution** ā If the upper limit of a class is the same as the lower limit of its next class then it is said to be exclusive or continuous Frequency Distribution

**Example**

# Draw the histogram of the following frequency distribution.

Daily earnings (in Rs) | 700 ā 750 | 750 ā 800 | 800 ā 850 | 850 ā 900 | 900 ā 950 | 950 ā 1000 |

No. of stores | 6 | 9 | 2 | 7 | 11 | 5 |

**Graphical Representation of Data**

# Since a picture represents better than a thousand words, The data is presented graphically. Some of the methods of representing the data graphically are:

**1. Bar Graph**

**2. Histogram**

**3. Frequency Polygon**

**1. Bar Graph**

# It is the easiest way to represent the data in the form of rectangular bars so it is called **Bar graph**.

- The thickness of each bar should be the same.
- The space between the bars should also be the same.
- The height of the bar should be according to the numerical data to be represented.

**Example**

# Represent the average monthly rainfall of Nepal for the first six months in the year 2014.

Month | Jan | Feb | Mar | Apr | May | Jun |

Average rainfall | 45 | 65 | 40 | 60 | 75 | 30 |

**Solution**

- On the x-axis mark the name of the months.
- On the y-axis mark the class interval which we have chosen.
- Then mark the average rainfall respective to the name of the month by the vertical bars.
- The bars could be of any width but should be the same.
- This is the required bar graph.

**2. Histogram**

# It is similar to Bar graph, but it is used in case of a continuous class interval.

- The class intervals are to be taken along an x-axis.
- The height represents the frequencies of the respective class intervals.

**Example**

# Draw the histogram of the following frequency distribution.

Daily earnings (in Rs) | 700 ā 750 | 750 ā 800 | 800 ā 850 | 850 ā 900 | 900 ā 950 | 950 ā 1000 |

No. of stores | 6 | 9 | 2 | 7 | 11 | 5 |

**Solution:**

- Mark the daily earnings on the x-axis.
- Mark the no. of stores on the y-axis.
- As the scale is starting from 700 so we will mark the zigzag to show the break.
- Mark the daily earnings through the vertical bars.

**3. Frequency Polygon**

**Procedure to draw the frequency polygon**

- First, we need to draw a histogram.
- Then join the midpoint of the top of the bars to a line segment and the figure so obtained is the required frequency polygon.
- The midpoint of the first bar is to be joined with the midpoint of the imaginary interval of the x-axis
- The midpoint of the last bar is to be joined with the midpoint of the next interval of the x-axis.

# If we need to draw the frequency polygon without drawing the histogram then first we need to calculate the class mark of each interval and these points will make the frequency polygon.

**Example**

# Draw the frequency polygon of a city in which the following weekly observations were made in a study on the cost of living index without histogram.

**Step 1:** First of all we need to calculate the class mark of each class interval.

**Step 2:** Take the suitable scale and represent the class marks on the x-axis.

**Step 3:** Take the suitable scale and represent the frequency distribution on the y-axis.

**Step 4:** To complete the frequency polygon we will join it with the x-axis before the first class and after the last interval.

**Step 5:** Now plot the respective points and join to make the frequency polygon.

**Measures of Central Tendency**

# To make all the study of data useful, we need to use measures of central tendencies. Some of the tendencies are

### 1. Mean

### 2. Median

### 3. Mode

**1. Mean (Average)**

# The mean is the average of the number of observations. It is calculated by dividing the sum of the values of the observations by the total number of observations.

# It is represented by x bar or.

# The meanof n values x1, x2, x3, …… xn is given by

**Mean of Grouped Data (Without Class Interval)**

# If the data is organized in such a way that the frequency is given but there is no class interval then we can calculate the mean by

# where, x1, x2, x3,…… xn are the **observations**

# f1, f2, f3, …… fn are the respective **frequencies** of the given observations.

**Example**

# Here, x1, x2, x3, x4, and x5 are 20, 40, 60, 80,100 respectively.

# and f1 , f2 , f3 , f4, f5 are 40, 60, 30, 50, 20 respectively.

**2. Median**

# The median is the middle value of the given number of the observation which divides into exactly two parts.

# For median of ungrouped data, we arrange it in ascending order and then calculated as follows

- If the number of the observations is odd then the median will be Term
- As in the above figure the no. of observations is 7 i.e. odd, so the median will beterm.

# = 4^{th} term.

# The fourth term is 44.

- If the number of observations is even then the median is the average of n/2 and (n/2) +1 term.

**Example**

# Find the median of the following data.

**6, 7, 10, 12, 13, 4, 8, 12**

# 1. First, we need to arrange it in ascending order.

**4, 6, 7,8,10,12,12,13**

# 2. The no. of observation is 8. As the no. of observation is even the median is the average of n/2 and (n/2)+1.

# 3.

# 4. 4^{th} term is 8 and the 5^{th} term is 10.

# 5. So the median

### 3. **Mode**

# The mode is the value of the observation which shows the number that occurs frequently in data i.e. the number of observations which has the maximum frequency is known as the Mode.

**Example**

# Find the Mode of the following data:

# 15, 20, 22, 25, 30, 20,15, 20,12, 20

**Solution**

# Here the number 20 appears the maximum number of times so

# Mode = 20.

# Remark: The empirical relation between the three measures of central tendency is: **3 Median = Mode + 2 Mean**

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