Statistics — Class 10 Maths Notes
Statistics · Class 10 Maths · 18 topics.
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Topics covered in Statistics
1.Introduction
Statistics in Simple Terms
Statistics is like a tool that helps us collect, analyze, interpret, and present data in a meaningful way. Imagine you have a lot of numbers or information (like scores in a cricket match or heights of students in your class), and you want to make sense of it all. Statistics helps with exactly that!
Basic Concepts and Formulas in Statistics
Mean (Average): This is what you usually call the average. You add up all the numbers and then divide by how many numbers there are. For example, if you have five numbers: 2, 4, 6, 8, 10, the mean is (2+4+6+8+10)/5 = 6.
Median: This is the middle value in a list of numbers. To find it, you arrange the numbers in order and then pick the middle one. If there's an even number of numbers, you take the average of the middle two. For instance, in the numbers 1, 3, 3, 6, 7, 8, 9, the median is 6.
Mode: This is the number that appears most often in a set. For example, in the numbers 1, 2, 2, 3, 3, 3, 4, the mode is 3 because it appears the most times.
Range: It's the difference between the highest and lowest numbers in a set. For example, in the numbers 5, 10, 15, 20, 25, the range is 25 - 5 = 20.
Where It's Used in Real Life and Careers
- In Daily Life: When you calculate your average marks in exams or when you see averages like batting averages in cricket.
- In Careers: Statistics is used in many fields like business (to understand market trends), medicine (for medical research), economics (to analyze economic data), and sports (to improve team performance).
Activity to Understand Better
Try this simple activity: Gather the ages of 10 of your friends or family members. Now calculate the mean, median, mode, and range of these ages. It's a fun way to see statistics in action!
2.Mean of Grouped Data
Mean of Grouped Data - Explained Simply
Happy Note: Finding the mean of grouped data is like being a chef who blends different ingredients to get the perfect taste for a dish. Each ingredient (data point) contributes to the overall flavor (mean)!
What is Mean of Grouped Data?
When data is grouped, like in classes or intervals, we calculate the mean to find the average value. This is especially useful when dealing with a large number of data points.
Formula for Mean of Grouped Data
The formula is: Mean=∑(×)∑Mean=∑f∑(f×x) where:
- f = Frequency of each group (how many times a data point occurs)
- x = Midpoint of each group (the average value of each interval)
- ∑∑ = Summation (adding up all the values)
Simple Example
Let's say we have ages of students grouped in intervals:
- 10-14 years: 5 students
- 15-19 years: 8 students
- 20-24 years: 3 students
First, find the midpoints:
- For 10-14, it's 10+142=12210+14=12
- For 15-19, it's 15+192=17215+19=17
- For 20-24, it's 20+242=22220+24=22
Then, multiply each midpoint by its frequency and add them up:
- (12×5)+(17×8)+(22×3)(12×5)+(17×8)+(22×3)
Finally, divide by total number of students:
- 60+136+665+8+35+8+360+136+66
This gives the mean age of the students.
Use in Real Life and Careers
In real life, this method is used in economics to understand income distribution, in meteorology to find average temperatures, and in education to analyze test scores. Career-wise, statisticians, economists, and data analysts frequently use this method for various analyses.
3.A survey was conducted by a group of students as a part of their environment awareness programme, in which they collected the following data regarding the number of plants in 20 houses in a locality. Find the mean number of plants per house.
The steps to calculate the mean are:
Find the Midpoint: Calculate the midpoint for each range of plants. This is done by adding the lower and upper limits of the range and dividing by 2.
Calculate the Product: Multiply the midpoint by the number of houses to find the total number of plants for that range.
Sum the Products: Add all the products from step 2 to find the total number of plants across all houses.
Sum the Frequencies: Add the number of houses to find the total number of houses surveyed.
Calculate the Mean: Divide the total number of plants by the total number of houses.
Here's the table with the values and the steps shown:
Now, we add up the 'f_i * x_i' column to get the total number of plants, and the 'Number of Houses (f_i)' column to get the total number of houses.
The formula for the mean is:
Mean=Total PlantsTotal HousesMean=Total HousesTotal Plants
Let's show the adding step and put the values in the formula to find the mean.
After adding up the total number of plants from all the ranges, we get 162 plants. The total number of houses surveyed is 20.
Using the formula for the mean:
Mean=Total PlantsTotal HousesMean=Total HousesTotal Plants
We substitute the values:
Mean=16220=8.1Mean=20162=8.1
So, the mean number of plants per house is 8.1.
4.Consider the following distribution of daily wages of 50 workers of a factory.(Find the mean daily wages of the workers of the factory by using an appropriate method.)
To find the mean daily wages of the factory workers from the provided distribution, we will follow these steps:
- Calculate the midpoint (class mark) of each wage group.
- Multiply each midpoint by the number of workers in that group to get the total wages for the group.
- Sum all the total wages from each group.
- Divide the sum of total wages by the total number of workers to get the mean wage.
Let's compute these steps and show the adding step and the formula application.
Here is the table that summarizes the data and calculations for finding the mean daily wages of the workers:
To calculate the mean daily wage, we add up the total wages from the 'f_i * x_i' column, which gives us 27,260. Since there are 50 workers in total, we use the formula:
Mean=Total WagesTotal Number of WorkersMean=Total Number of WorkersTotal Wages
Substituting the values into the formula:
Mean=2726050=545.2Mean=5027260=545.2
Therefore, the mean daily wages of the workers in the factory is ₹545.20.
5.Thirty women were examined in a hospital by a doctor and the number of heartbeats per minute were recorded and summarised as follows. Find the mean heartbeats per minute for these women, choosing a suitable method.
To calculate the mean heartbeats per minute for the women, we have made a table with the given data and performed the following steps:
Determine the Midpoint: For each range of heartbeats per minute, we found the midpoint by averaging the lower and upper limits. For example, the midpoint for the 65-68 range is (65+68)/2=66.5(65+68)/2=66.5.
Calculate Total Heartbeats: We then multiplied each midpoint by the number of women in that range to get the total heartbeats. For example, for the 65-68 range, the total heartbeats is 66.5×2=13366.5×2=133.
Sum of Total Heartbeats: We added all the total heartbeats to find the total number of heartbeats for all women.
Calculate the Mean: The mean is calculated by dividing the sum of total heartbeats by the total number of women.
Here's the detailed table with the calculations:
The sum of total heartbeats is 2277.0. Since there are 30 women, the mean heartbeats per minute is:
Mean heartbeats per minute=2277.030=75.9Mean heartbeats per minute=302277.0=75.9
Therefore, the mean heartbeats per minute for these women is 75.9. This is the average number of heartbeats per minute for the group of women examined
6.To find out the concentration of SO2 in the air (in parts per million, i.e., ppm), the data was collected for 30 localities in a certain city and is presented below: Find the mean concentration of SO2 in the air.
The mean concentration of 2SO2 in the air for the 30 localities is approximately 0.099 ppm (parts per million).
Here is the table that shows the calculation details:
To calculate the mean, we sum up the products of the midpoints and frequencies (column "f_i * x_i") and then divide by the total frequency (which is the total number of localities). The calculation is as follows:
Mean concentration=Total of (∗)Total frequency=2.9630≈0.099 ppmMean concentration=Total frequencyTotal of (fi∗xi)=302.96≈0.099 ppm
7.A class teacher has the following absentee record of 40 students of a class for the whole term. Find the mean number of days a student was absent.
The mean number of days a student was absent is approximately 12.48 days.
Here is the table that shows the calculation details:
To calculate the mean, we sum up the products of the midpoints and frequencies (column "f_i * x_i") and then divide by the total number of students (40). The calculation is as follows:
Mean number of absent days=Total of (∗)Total number of students=499.040≈12.48Mean number of absent days=Total number of studentsTotal of (fi∗xi)=40499.0≈12.48
8.The following table gives the literacy rate (in percentage) of 35 cities. Find the mean literacy rate.
The mean literacy rate for the 35 cities is approximately 69.43%.
Here is the table that shows the calculation details:
To calculate the mean literacy rate, we perform the following steps:
- Find the Midpoint: For each interval of literacy rates, find the midpoint by calculating the average of the lower and upper bounds.
- Multiply by Frequency: Multiply the midpoint by the number of cities to get the total literacy rate for that interval.
- Sum of Total Literacy Rates: Add all the total literacy rates from each interval.
- Divide by Total Cities: Divide the sum of total literacy rates by the total number of cities to find the mean.
Using the formula for the mean:
Mean literacy rate=Total of (∗)Total number of citiesMean literacy rate=Total number of citiesTotal of (fi∗xi)
The calculation is as follows:
Mean literacy rate=2430.035≈69.43%Mean literacy rate=352430.0≈69.43%
This is the average literacy rate across all 35 cities.
9.Mode of Grouped Data
The mode is a type of average, specifically the value that appears most frequently in a data set. When dealing with grouped data, however, things are a bit different than with individual data points.
Understanding Mode in Grouped Data
Grouped Data: First, grouped data means your data is divided into intervals (or groups), not just listed as individual values.
Identifying the Modal Group: To find the mode in grouped data, you need to identify which group (or interval) has the highest frequency, meaning the group where the most data points fall.
Formula for Finding Mode: While the exact value of the mode in grouped data can't always be pinpointed, we can estimate it using a formula. The formula is:
Mode=+(−2−−)×ℎMode=L+(2fm−fp−fnfm−fp)×h
Here:
- L is the lower boundary of the modal group.
- fm is the frequency of the modal group.
- fp is the frequency of the group before the modal group.
- fn is the frequency of the group after the modal group.
- ℎh is the width of the groups (assumed to be the same for all groups).
Real-Life Example and Application
Imagine you have a survey of the number of books people read in a month, and you group the data like this: 0-4 books, 5-9 books, 10-14 books, etc. If most people fall into the 5-9 books category, that's your modal group. By applying the formula, you can estimate the most common number of books read.
Application in Careers and Industries
Understanding the mode of grouped data is particularly useful in fields like market research, statistics, and data analysis. For example, a company may use this method to determine the most popular product size or price range among its customers. This information is crucial for making informed business decisions, like planning inventory or setting prices
10.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.
To find the mode of the given data, we first create a table showing the frequency of each number of wickets taken in the 10 cricket matches. Here's the table:
Now, to find the mode, we look for the number of wickets that was taken most frequently. In this case, the bowler took 2 wickets in 3 matches, which is more frequent than any other number of wickets taken. Therefore, the mode of this data is 2 wickets.11.The following data gives the information on the observed lifetimes (in hours) of 225 electrical components : (Determine the modal lifetimes of the components.)
The modal monthly expenditure of the families is found by identifying the expenditure range with the highest frequency. The mean monthly expenditure is calculated using the formula:
Mean=∑(Mid-Point×Frequency)Total Number of FamiliesMean=Total Number of Families∑(Mid-Point×Frequency)
Here's the calculation in a tabular format:
Expenditure Range Number of Families Mid-Point Weighted Expenditure 1000-1500 24 1250 30000 1500-2000 40 1750 70000 2000-2500 33 2250 74250 2500-3000 28 2750 77000 3000-3500 30 3250 97500 3500-4000 22 3750 82500 4000-4500 16 4250 68000 4500-5000 7 4750 33250 Total / Mean Mean / Mode 2662.5 532500 The modal monthly expenditure range is ₹1500-2000, with the highest frequency of 40 families.
The mean monthly expenditure is calculated as follows: Mean=532500200=₹2662.5Mean=200532500=₹2662.5
Therefore, the mean monthly expenditure per family is approximately ₹2662.5. This gives us an average expenditure across all families, whereas the modal expenditure range gives us the most common expenditure range among the families.
12.Median of Grouped Data
What is the Median?
The median is the middle value of a dataset. When the data is arranged in order, it's the value that splits the data into two equal halves.
Median of Grouped Data
When dealing with grouped data, we can't find the exact median because we don't know the exact values, just the intervals they fall into. But we can estimate it.
How to Find the Median in Grouped Data
- Arrange the Data in Ascending Order: This is usually done for you in a frequency table.
- Find the Cumulative Frequency: Add up the frequencies as you move down the table. This shows the total number of observations up to that point.
- Locate the Median Class: This is the group where the median lies. We find it by looking for the class where the cumulative frequency first equals or exceeds half the total number of observations.
- Use the Formula: Median=+(2−)×Median=L+(f2N−F)×w
- L = Lower boundary of the median class.
- N = Total number of observations.
- F = Cumulative frequency of the class before the median class.
- f = Frequency of the median class.
- w = Width of the median class.
Example
Suppose you have the following frequency table of scores:
Score Range Frequency 0-20 5 21-40 10 41-60 8 61-80 7 81-100 4 Total observations (N) = 34. Half of this is 17. The cumulative frequency just greater than 17 is in the 41-60 range. Using the formula, you can calculate the median.
Real-Life Application
Understanding the median of grouped data is helpful in various fields like economics, sociology, and environmental science. It helps to understand the central tendency of large sets of data, which is crucial in these fields for making informed decisions.
Activity
Try to find the median of different datasets on your own. You can use data from sports, like cricket scores, or exam marks of your class.
13.If the median of a distribution given below is 28.5, find the value of x & y.
Solution:
Given data, n = 60
Median of the given data = 28.5
Where, N/2 = 30
Median class is 20 – 30 with a cumulative frequency = 25 + x
Lower limit of median class, l = 20,
cf = 5 + x,
f = 20 & h = 10
Median=l+(f2N−F)×h
Median=+2−×ℎSubstitute the values
28.5 = 20 + [(30 − 5 − x)/20] × 10
8.5 = (25 – x)/2
17 = 25 – x
Therefore, x = 8.
Now, from cumulative frequency, we can identify the value of x + y as follows:
Since,
60 = 45 + x + y
Now, substitute the value of x, to find y
60 = 45 + 8 + y
y = 60 – 53
y = 7
Therefore, the value of x = 8 and y = 7.
14.The life insurance agent found the following data for the distribution of ages of 100 policy holders. Calculate the median age, if policies are given only to the persons whose age is 18 years onwards but less than the 60 years.
Solution:Given data: N = 100 and N/2 = 50
Median class = 35-40
Then, l = 35, cf = 45, f = 33 & h = 5
Median=l+(f2N−F)×hMedian = 35 + [(50 – 45)/33] × 5
= 35 + (25/33)
= 35.76
Therefore, the median age = 35.76 years.
15.The lengths of 40 leaves in a plant are measured correctly to the nearest millimeter, and the data obtained is represented as in the following table: (Find the median length of the leaves. (Hint : The data needs to be converted to continuous classes for
Solution:-Since the data are not continuous reduce 0.5 in the lower limit and add 0.5 in the upper limit.
So, the data obtained are:
N = 40 and N/2 = 20
Median class = 144.5-153.5
then, l = 144.5,
cf = 17, f = 12 & h = 9
Median=l+(f2N−F)×hMedian = 144.5 + [(20 – 17)/ 12] × 9
= 144.5 + (9/4)
= 146.75 mm
Therefore, the median length of the leaves = 146.75 mm.
16.100 surnames were randomly picked up from a local telephone directory and the frequency distribution of the number of letters in the English alphabets in the surnames was obtained as follows:Determine the median number of letters in the surnames. Find the
Solution::To calculate median:
Given:
N = 100 & N/2 = 50
Median class = 7-10
Therefore, l = 7, cf = 36, f = 40 & h = 3
Median=l+(f2N−F)×hMedian = 7 + [(50 – 36)/40] × 3
Median = 7 + (42/40)
Median = 8.05
Calculate the Mode:
Modal class = 7-10,
Where, l = 7, f1 = 40, f0 = 30, f2 = 16 & h = 3
Mode = 7 + [(40 – 30)/(2 × 40 – 30 – 16)] × 3
= 7 + (30/34)
= 7.88
Therefore mode = 7.88
Calculate the Mean:
Mean = x̄ = ∑fi xi /∑fi
Mean = 832/100 = 8.32
Therefore, mean = 8.32
17.The distribution below gives the weights of 30 students of a class. Find the median weight of the students.
Solution:
Given: N = 30 and N/2= 15
Median class = 55-60
l = 55, Cf = 13, f = 6 & h = 5
Median=l+(f2N−F)×hMedian = 55 + [(15 – 13)/6] × 5
= 55 + (10/6)
= 55 + 1.666
= 56.67
Therefore, the median weight of the students = 56.67
18.Quick Revision
1. Introduction of Statistics
Statistics is a part of math that deals with collecting, analyzing, interpreting, and presenting data. It helps us make sense of numbers and find patterns in them.
2. Mean of Grouped Data
The mean is the average. For grouped data, you find the mean by multiplying the midpoint of each group by the number of items in that group, adding all these products together, and then dividing by the total number of items. Formula: Mean=∑(×)∑Mean=∑f∑(f×x), where f is the frequency and x is the midpoint.
3. Mode of Grouped Data
The mode is the most common value. For grouped data, it's the value where items occur most often. It's estimated using the formula: Mode=+(−−1)(−−1)+(−+1)×ℎMode=L+(fm−fm−1)+(fm−fm+1)(fm−fm−1)×h, where L is the lower boundary of the modal class, fm is the frequency of the modal class, −1fm−1 and +1fm+1 are frequencies of the classes before and after the modal class, and ℎh is the class width.
4. Median of Grouped Data
The median is the middle value. In grouped data, it's calculated by finding the class where the median lies and using the formula: Median=+(2−)×ℎMedian=L+f(2n−F)×h, where n is the total number of items, F is the cumulative frequency before the median class, f is the frequency of the median class, and ℎh is the class width.