Statistics

Introduction

Statistics is a branch of mathematics that deals with the collection, organization, presentation, analysis, and interpretation of data. It helps us to make decisions based on data collected from various sources. In our daily life, we encounter data in different forms such as marks obtained by students, daily temperature, number of plants in a locality, etc. The process of statistics involves several steps: first, data collection from a population or sample; second, organizing the data in a meaningful way; third, presenting the data using tables, graphs, or charts; fourth, analyzing the data to find patterns or trends; and finally, interpreting the results to make informed decisions. This chapter focuses on the measures of central tendency, which summarize data by identifying a central or typical value. The three main measures of central tendency discussed are mean, median, and mode. These help us understand the average or most common values in the data set. Understanding these concepts is essential for analyzing data effectively in various fields such as education, economics, and social sciences. **Table on page 8 (1×3)** | Percentage of female teachers | Number of States /U.T. ( f ) i | x i | | --- | --- | --- | | 15 - 25 25 - 35 35 - 45 45 - 55 55 - 65 65 - 75 75 - 85 | 6 11 7 4 4 2 1 | 20 30 40 50 60 70 80 | **Table on page 11 (1×8)** | Number of plants | 0 - 2 | 2 - 4 | 4 - 6 | 6 - 8 | 8 - 10 | 10 - 12 | 12 - 14 | | --- | --- | --- | --- | --- | --- | --- | --- | | Number of houses | 1 | 2 | 1 | 5 | 6 | 2 | 3 | **Table on page 12 (4×13)** | | Number of heartbeats per minute | | 65 - 68 | 68 - 71 | 71 - 74 | | 74 - 77 | 77 - 80 | 80 - 83 | | 83 - 86 | | | --- | --- | --- | --- | --- | --- | --- | --- | --- | --- | --- | --- | --- | | Number of women 2 4 3 8 7 4 2 5. In a retail market, fruit vendors were selling mangoes kept in packing boxes. These boxes contained varying number of mangoes. The following was the distribution of mangoes according to the number of boxes. Number of mangoes 50 - 52 53 - 55 56 - 58 59 - 61 62 - 64 Number of boxes 15 110 135 115 25 Find the mean number of mangoes kept in a packing box. Which method of finding the mean did you choose? 6. The table below shows the daily expenditure on food of 25 households in a locality. Daily expenditure 100 - 150 150 - 200 200 - 250 250 - 300 300 - 350 (in `) Number of 4 5 12 2 2 households Find the mean daily expenditure on food by a suitable method. 7. To find out the concentration of SO in the air (in parts per million, i.e., ppm), the data 2 was collected for 30 localities in a certain city and is presented below: Concentration of SO (in ppm) Frequency 2 0.00 - 0.04 4 0.04 - 0.08 9 0.08 - 0.12 9 0.12 - 0.16 2 0.16 - 0.20 4 | Number of women | | 2 | 4 | 3 | | 8 | 7 | 4 | | 2 | | | | | Concentration of SO (in ppm) 2 | | | | Frequency | | | | | | | | | | 0.00 - 0.04 0.04 - 0.08 0.08 - 0.12 0.12 - 0.16 0.16 - 0.20 | | | | 4 9 9 2 4 | | | | | | | | | | 0.20 - 0.24 | | | | 2 | | | | | | | **Table on page 13 (5×18)** | | | Number of days | | 0 - 6 | | 6 - 10 | 10 - 14 | 14 - 20 | | 20 - 28 | | 28 - 38 | | 38 - 40 | | | | | --- | --- | --- | --- | --- | --- | --- | --- | --- | --- | --- | --- | --- | --- | --- | --- | --- | --- | | | | Number of | | 11 | | 10 | 7 | 4 | | 4 | | 3 | | 1 | | | | | students 9. The following table gives the literacy rate (in percentage) of 35 cities. Find the mean literacy rate. Literacy rate (in %) 45 - 55 55 - 65 65 - 75 75 - 85 85 - 95 Number of cities 3 10 11 8 3 13.3 Mode of Grouped Data Recall from Class IX, a mode is that value among the observations which occurs most often, that is, the value of the observation having the maximum frequency. Further, we discussed finding the mode of ungrouped data. Here, we shall discuss ways of obtaining a mode of grouped data. It is possible that more than one value may have the same maximum frequency. In such situations, the data is said to be multimodal. Though grouped data can also be multimodal, we shall restrict ourselves to problems having a single mode only. Let us first recall how we found the mode for ungrouped data through the following example. Example 4 : The wickets taken by a bowler in 10 cricket matches are as follows: 2 6 4 5 0 2 1 3 2 3 Find the mode of the data. Solution : Let us form the frequency distribution table of the given data as follows: Number of 0 1 2 3 4 5 6 wickets | | students | | | | | | | | | | | | | | | | | | Number of wickets | | 0 | | 1 | | 2 | | 3 | | 4 | | 5 | | 6 | | | | | | | | | | | | | | | | | | | | | | | | Number of matches | | 1 | | 1 | | 3 | | 2 | | 1 | | 1 | | 1 | | |

• Statistics involves collecting, organizing, presenting, analyzing, and interpreting data.
• Data can be collected from populations or samples.
• Measures of central tendency summarize data by identifying a central value.
• Mean, median, and mode are the three main measures of central tendency.
• Statistics helps in making informed decisions based on data.
• Data presentation includes tables, graphs, and charts.
• 📌 Statistics: Branch of mathematics dealing with data collection and analysis.
• 📌 Data: Collection of observations or measurements.
• 📌 Measures of Central Tendency: Values that represent the center of a data set.

Mean of Ungrouped Data

The mean or average of ungrouped data is calculated by summing all the observations and dividing by the total number of observations. Suppose we have observations x1, x2, ..., xn with corresponding frequencies f1, f2, ..., fn. The total number of observations is the sum of all frequencies (f1 + f2 + ... + fn), and the sum of the values of all observations is (f1×x1 + f2×x2 + ... + fn×xn). The mean (x̄) is given by the formula: x̄ = (Σfi×xi) / (Σfi). This method is straightforward when the data is ungrouped, meaning individual data points are known. For example, if marks obtained by students are listed individually, their mean can be calculated by adding all marks and dividing by the number of students. The mean gives a measure of the central value of the data set and is sensitive to extreme values (outliers). **Table on page 2 (4×5)** |  f 1 to n. i Let us apply this formula to find the mean in the following example. Example 1 : The marks obtained by 30 students of Class X of a certain school in a Mathematics paper consisting of 100 marks are presented in table below. Find the mean of the marks obtained by the students. Marks obtained 10 20 36 40 50 56 60 70 72 80 88 92 95 (x) i Number of 1 1 3 4 3 2 4 4 1 1 2 3 1 students ( f) i Solution: Recall that to find the mean marks, we require the product of each x with i the corresponding frequency f. So, let us put them in a column as shown in Table 13.1. i Table 13.1 Marks obtained (x ) Number of students ( f ) fx i i i i 10 1 10 20 1 20 . 36 3 108 40 4 160 50 3 150 56 2 112 60 4 240 70 4 280 72 1 72 80 1 80 88 2 176 | | | | | | --- | --- | --- | --- | --- | | | Marks obtained (x ) i | Number of students ( f ) i | fx i i | | | | 10 20 . 36 40 50 56 60 70 72 80 88 | 1 1 3 4 3 2 4 4 1 1 2 | 10 20 108 160 150 112 240 280 72 80 176 | | | | 92 95 | 3 1 | 276 95 | | | | Total | f = 30 i | fx = 1779 i i | | **Table on page 2 (1×14)** | Marks obtained (x) i | 10 | 20 | 36 | 40 | 50 | 56 | 60 | 70 | 72 | 80 | 88 | 92 | 95 | | --- | --- | --- | --- | --- | --- | --- | --- | --- | --- | --- | --- | --- | --- | | Number of students ( f) i | 1 | 1 | 3 | 4 | 3 | 2 | 4 | 4 | 1 | 1 | 2 | 3 | 1 |

• Mean is the sum of all observations divided by the number of observations.
• For ungrouped data, individual values are known.
• Frequencies represent how many times each observation occurs.
• Mean formula: x̄ = (Σfi×xi) / (Σfi).
• Mean is sensitive to extreme values.
• Used to find the average value in a data set.
• 📌 Mean (x̄): Average value of data set.
• 📌 Frequency (fi): Number of times an observation occurs.