## CBSE Class 10 Maths Notes Chapter 14 Statistics

# CBSE Class 10 Maths Chapter 14: Statistics

## Introduction

- Statistics is a branch of mathematics that deals with the collection, analysis, interpretation, presentation, and organization of data.

## Mode

- Mode is the value that appears most frequently in a dataset.
- A dataset may have one mode, more than one mode, or no mode.
- The mode of a dataset can be found by arranging the data in ascending or descending order and then identifying the value that appears most frequently.

## Median

- Median is the middle value of a dataset when the data is arranged in order.
- The median is useful when the dataset contains extreme values or outliers that may skew the mean.
- To find the median, arrange the data in ascending or descending order and then identify the middle value.

## Mean

- Mean is the average value of a dataset.
- To find the mean, add up all the values in the dataset and then divide the sum by the number of values in the dataset.

## Range

- Range is the difference between the largest and smallest values in a dataset.
- To find the range, subtract the smallest value from the largest value.

## Quartiles

- Quartiles are values that divide a dataset into four equal parts.
- The first quartile (Q1) is the value that separates the lowest 25% of the data from the rest of the data.
- The second quartile (Q2) is the same as the median.
- The third quartile (Q3) is the value that separates the highest 25% of the data from the rest of the data.

## Interquartile Range

- The interquartile range is the difference between the third quartile (Q3) and the first quartile (Q1).
- The interquartile range provides a measure of the spread or dispersion of the middle 50% of the data.

## Outliers

- Outliers are values that are significantly different from the rest of the values in a dataset.
- Outliers can significantly affect the mean and standard deviation of a dataset.
- It is important to identify and understand outliers to avoid misinterpreting statistical results.

## Frequency Distribution

- Frequency distribution is a table that shows the number of times each value appears in a dataset.
- A frequency distribution can help to identify the mode, median, and range of a dataset.

## Cumulative Frequency Distribution

- Cumulative frequency distribution is a table that shows the total number of values that fall below a certain value in a dataset.
- A cumulative frequency distribution can help to identify quartiles and percentiles of a dataset.

## Histogram

- Histogram is a graphical representation of a frequency distribution.
- A histogram consists of bars that represent the frequency or proportion of values in each interval.
- A histogram can help to visualize the shape, center, and spread of a dataset.

## Pie Chart

- A pie chart is a graphical representation of data that uses a circle to represent the whole and slices to represent parts of the whole.
- The size of each slice is proportional to the quantity it represents.

## Bar Graph

- A bar graph is a graphical representation of data that uses bars to represent the quantities or values of different categories.
- The length of each bar is proportional to the value it represents.

## Measures of Central Tendency

- Measures of central tendency are statistical measures that indicate where the center of the data lies.
- The most common measures of central tendency are the mean, median, and mode.
- The mean is the most commonly used measure of central tendency, but it can be affected by outliers.

## Measures of Dispersion

- Measures of dispersion are statistical measures that indicate

**CBSE Class 10 Maths Important Questions Chapter 14 Statistics**

- Define the range of a data set.
- What is the difference between mean and median?
- What is the mode of a data set?
- What is a frequency distribution table?
- Define the interquartile range.
- What is a cumulative frequency distribution table?
- How is the mean deviation of grouped data calculated?
- What is the difference between a histogram and a bar graph?
- Define the coefficient of variation.
- What is a pie chart?
- What is a box plot?
- Define the standard deviation of a data set.
- What is the purpose of measures of central tendency?
- What is the purpose of measures of dispersion?
- Define the empirical rule.

**CBSE Class 10 Maths Important Questions Answers Chapter 14 Statistics**

- What is the median of the following data: 3, 4, 2, 1, 7, 5, 6, 4, 8?

Answer: Arranging the given data in ascending order, we get: 1, 2, 3, 4, 4, 5, 6, 7, 8

The median of the given data is the middle value. As there are 9 values in total, the middle value will be the 5th value.

Therefore, the median of the given data is 4.

- Find the mean of the following data: 15, 20, 25, 30, 35.

Answer: To find the mean of the given data, we need to add up all the values and then divide by the total number of values.

Adding the given values, we get: 15 + 20 + 25 + 30 + 35 = 125

There are 5 values in total.

So, the mean of the given data is: 125/5 = 25

Therefore, the mean of the given data is 25.

- The mean and median of 10 observations are 7 and 8 respectively. Find the value of the observation which occurs twice.

Answer: Let the observation which occurs twice be x.

The total sum of the given observations can be calculated as follows:

Mean = (sum of all the observations) / (total number of observations) 7 = (sum of all the observations) / 10 sum of all the observations = 70

Since the median is 8, we know that the 5th and 6th observations are both x.

Also, since x occurs twice, the sum of the remaining 8 observations will be 70 – 2x.

Therefore, we have:

x + x + sum of the remaining 8 observations = 70 2x + sum of the remaining 8 observations = 70 sum of the remaining 8 observations = 70 – 2x

Since the median of the given observations is 8, we know that the sum of the first 4 observations is equal to the sum of the last 4 observations.

Therefore, we have:

(sum of the first 4 observations) = (sum of the last 4 observations) (1 + 2 + 3 + 4) + (5 + 6 + 7 + x) = (x + 7 + 8 + 9) + 10 10 + x + 15 = x + 34 x = 9

Therefore, the value of the observation which occurs twice is 9.

- The following table shows the marks obtained by a group of students in a mathematics test. Find the mean, median and mode of the given data.

Marks | No. of Students |
---|---|

20 | 3 |

25 | 6 |

30 | 10 |

35 | 8 |

40 | 3 |

Answer: To find the mean of the given data, we need to calculate the weighted average of the marks, where the weights are given by the number of students who obtained each mark.

Mean = (20×3 + 25×6 + 30×10 + 35×8 + 40×3) / (3 + 6 + 10 + 8 + 3) = 30

To find the median, we need to arrange the given data in ascending order and then find the middle value.

20, 20, 20, 25, 25, 25,

**CBSE Class 10 Maths Important Questions Answers MCQs Chapter 14 Statistics**

- In a class of 40 students, the heights (in cm) of 10 students are as follows: 150, 155, 162, 147, 160, 155, 148, 162, 153, 149. What is the mode of the heights?

A. 155 B. 162 C. 148 D. 149

Answer: A. 155

Mode is the value that occurs most frequently in a given data set. In the given data set, the height 155 occurs twice, which is more frequent than any other height. Therefore, the mode of the heights is 155.

- The marks obtained by 30 students in a mathematics test are as follows: 40, 45, 50, 55, 60, 65, 70, 75, 80, 85, 90, 95, 100, 105, 110, 115, 120, 125, 130, 135, 140, 145, 150, 155, 160, 165, 170, 175, 180. What is the median of the marks?

A. 85 B. 90 C. 95 D. 100

Answer: C. 95

Median is the middle value in a data set when arranged in ascending or descending order. In the given data set, there are 30 values, so the median will be the 15th value when arranged in ascending order. The 15th value in the data set is 95, which is the median of the marks.

- The following table shows the number of hours spent by a group of students in a week on different activities. What is the mode of the data?

Activity | No. of Students |
---|---|

Study | 12 |

Sports | 8 |

TV | 10 |

Outdoor Fun | 5 |

Others | 5 |

A. Study B. Sports C. TV D. Outdoor Fun

Answer: A. Study

Mode is the value that occurs most frequently in a given data set. In the given data set, the activity “Study” has the highest frequency, with 12 students spending time on it, which is more frequent than any other activity. Therefore, the mode of the data is “Study”.

- The marks obtained by a class of 50 students in a mathematics test are as follows: 40, 45, 50, 55, 60, 65, 70, 75, 80, 85, 90, 95, 100, 105, 110, 115, 120, 125, 130, 135, 140, 145, 150, 155, 160, 165, 170, 175, 180. What is the mean of the marks?

A. 87 B. 90 C. 93 D. 95

Answer: C. 93

Mean is the average of a data set, which is calculated by summing up all the values and dividing by the total number of values. In the given data set, the sum of all the marks is 4650. Dividing this sum by the total number of students (50), we get the mean as 93. Therefore, the mean of the marks is 93.

## CBSE Class 10 Maths Notes Chapter 14 Statistics

**MEAN (AVERAGE):** Mean [Ungrouped Data] – Mean of n observations, x_{1}, x_{2}, x_{3} … x_{n}, is

**MEAN [Grouped Data]:** The mean for grouped data can be found by the following three methods:**(i) Direct Mean Method:**

Class Mark = UpperClassLimit+LowerClassLimit2

Note: Frequency of a class is centred at its mid-point called class mark.

**(ii) Assumed Mean Method:** In this, an arbitrary mean ‘a’ is chosen which is called, ‘assumed mean’, somewhere in the middle of all the values of x.

…[where d_{i} = (x_{i} – a)]

**(iii) Step Deviation Method:**

….. [where ui=dih , where h is a common divisor of d_{i}]

**MEDIAN:** Median is a measure of central tendency that gives the value of the middle-most observation in the data.

…where[l = Lower limit of median class; n = Number of observations; f = Frequency of median class; c.f. = Cumulative frequency of preceding class; h = Class size]

(iii) Representing a cumulative frequency distribution graphically as a cumulative frequency curve, or an ogive of the less than type and of the more than type. The median of grouped data can be obtained graphically as the x-coordinate of the point of intersection of the two ogives for this data.

**Mode:**

(i) Ungrouped Data: The value of the observation having maximum frequency is the mode.

(ii) Grouped Data:

…where[l = Lower limit of modal class; f_{1} = Frequency of modal class; f_{0} = Frequency of the class preceding the modal class; f_{2} = Frequency of the class succeeding the modal class; h = Size of class interval. c.f. = Cumulative frequency of preceding class; h = Class size]

Mode = 3 Median – 2 Mean

Median = Mode+2Mean3

Mean = 3Median−Mode2

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