## Revision Notes for Ch 14 Statistics Class 10th Mathematics

**Concept of Mean, Median and Mode**

• Average or mean is the certain value representative of the whole data and signifying its characteristics is called average of the data. It is also known as mean.

• The average is given as

• Median of data is central data, obtained when data is arranged in ascending order. Mode is the data, having maximum frequency.

• Mode is data having maximum Frequency.

**Grouped Data**

• Presenting a large number of data, we need to group data for certain range, and get the frequency of that group.

Marks Obtained | Frequency |
---|---|

0-4 | 3 |

4-9 | 1 |

10-14 | 7 |

15-19 | 3 |

20-24 | 6 |

Here, frequency of class Size is 0-4 is 3. It means 3 students has obtained marks in the range of 0-4.

• Above is the grouped the data, having range of 4 marks, i.e 0-4, 4-9.These groupings are called ‘classes’ or ‘class-intervals’, and their size is called the class-size or class width.

• Class width is chosen according convenience.

**Consider the following Grouped Data**

Class | Frequency |
---|---|

0-10 | 2 |

10-20 | 4 |

20-30 | 5 |

30-40 | 3 |

In class size, if there is 10 as one of the observation, it is not represented in 0-10. It is represented in 10-20. Similarly, If 20 is one of the data; it is included in 20-30. Not in 10-20. So,

__upper limit is included, and lower limit is excluded.__

__Inclusive Grouped Data__In this, both upper limit and lower limit is included.

**Consider the following Grouped Data**

That is in class 0-9, both 9 is also included.

Class | Frequency |
---|---|

0-9 | 3 |

10-19 | 5 |

20-29 | 4 |

30-39 | 3 |

That is in class 0-9, both 9 is also included.

**Direct Method for Finding Mean**

**Assumed Mean Method**

**Mean of Inclusive Data**

• To find mean of inclusive series, it is first needed to be converted into exclusive series.

• It is done, by subtracting 0.5 from upper and adding 0.5 to lower limit. After that, you can calculate mean by whatever method you like, or by the method which is mentioned in question.

• The class size will be 1 more for exclusive series for corresponding inclusive series.

**Note:**After converting Inclusive series in exclusive series, we can find mean, median and mode of the given data.

__Example:__

Class |
15-19 | 20-24 | 25-29 | 30-34 | 35-39 | 40-44 | 45-49 |

Frequency |
3 | 13 | 21 | 15 | 5 | 4 | 2 |

So, here we have inclusive series, We will convert it into exclusive like below

Class | 15.5-19.5 | 20.5-24.5 | 25.5-29.5 | 30.5-34.5 | 35.5-39.5 | 40.5-44.5 | 45.5-49.5 |

Frequency | 3 | 13 | 21 | 15 | 5 | 4 | 2 |

**Cumulative Frequency**

• Cumulative frequency is the running total of all frequencies.

**Consider the following Data**

• The following data represent the marks obtained by 100 students in science test.

Marks |
Frequency(Number of Students) |
Cumulative Frequency |

0-10 | 5 | 5 |

10-20 | 20 | 25 |

20-30 | 45 | 70 |

30-40 | 20 | 90 |

40-50 | 10 | 100 |

As you can see that, cumulative frequency is the sum of all frequencies of all class size, till that class.

__What Cumulative frequency signifies?__

Marks less than |
Number of students |

10 | 5 |

20 | 25 |

30 | 70 |

40 | 90 |

50 | 100 |

So, by using cumulative frequency, we get that, how many students scored less than a particular mark. Similarly, we can represent the above data, in following form.

__Less Than__

Marks less than | Number of students |

10 | 5 |

20 | 25 |

30 | 70 |

40 | 90 |

50 | 100 |

__More Than__

Marks less than | Number of students |

0 | 100 |

10 | 95 |

20 | 75 |

30 | 30 |

40 | 10 |

50 | 0 |

**Median**

**Mode**

**Mode= 3(median)-2(mean)**

**Cumulative Frequency Curve (Ogive)**

**Tabulate the ‘less than’ and ‘more than’ cumulative series, as mentioned earlier.**

• On a graph paper, we mark the upper class limits along X axis and corresponding cumulative frequencies along y axis.

• Take a point A(0,N/2) on the y axis and draw AP II X-axis, cutting the above curve at a point P. Draw PM perpendicular to x-axis.

• We can draw it for both greater than and lower than series.

• The point as much both curve intersect is median.

• The curve obtained by less than series is known as less than ogive. The curve obtained by more than series is known as more than Ogive.

**Q. Draw less than and more than Ogive of the following data, and find the median.**

Capital (In lakh of Rs) |
Number of Companies |

0-10 | 2 |

10-20 | 3 |

20-30 | 7 |

30-40 | 11 |

40-50 | 15 |

50-60 | 7 |

60-70 | 2 |

70-80 | 3 |

**Solution:**

First we will find sum of all frequencies.

Now, we will draw a more than series table of the above data.

Capital (In lakh of Rs) | Number of Companies |

0 | 50 |

10 | 48 |

20 | 45 |

30 | 38 |

40 | 27 |

50 | 12 |

60 | 5 |

70 | 3 |

80 | 0 |

Now, we will draw less.

Capital (In lakh of Rs) | Number of Companies |

0 | 0 |

10 | 2 |

20 | 5 |

30 | 12 |

40 | 23 |

50 | 38 |

60 | 45 |

70 | 47 |

80 | 50 |

Now, we will plot Graphs for both, one by one.