This document discusses measures of central tendency and variation for numerical data. It defines and provides formulas for the mean, median, mode, range, variance, standard deviation, and coefficient of variation. Quartiles and interquartile range are introduced as measures of spread less influenced by outliers. The relationship between these measures and the shape of a distribution are also covered at a high level.
This document discusses measures of dispersion and the normal distribution. It defines measures of dispersion as ways to quantify the variability in a data set beyond measures of central tendency like mean, median, and mode. The key measures discussed are range, quartile deviation, mean deviation, and standard deviation. It provides formulas and examples for calculating each measure. The document then explains the normal distribution as a theoretical probability distribution important in statistics. It outlines the characteristics of the normal curve and provides examples of using the normal distribution and calculating z-scores.
1. Measures of central tendency include the mean, median, and mode.
2. The mean is the average value found by dividing the sum of all values by the total number of values. The median is the middle value when values are arranged in order. The mode is the value that appears most frequently.
3. For grouped data, the mean is calculated using the sum of the frequency multiplied by the class midpoint divided by the total frequency. The median class is identified which has a cumulative frequency above and below half the total. The mode is the class with the highest frequency.
This document provides information on measures of central tendency and dispersion. It discusses the mean, median, and mode as the three main measures of central tendency. It provides formulas and examples for calculating the mean, median, and mode for both ungrouped and grouped data. The document also covers measures of dispersion including range, semi-interquartile range, variance, standard deviation, and coefficient of variation. It provides formulas and examples for calculating each of these measures. Finally, the document briefly discusses chi-square tests, Pearson's correlation, and using scatterplots to examine relationships between variables.
Unit-I Measures of Dispersion- Biostatistics - Ravinandan A P.pdfRavinandan A P
Biostatistics, Unit-I, Measures of Dispersion, Dispersion
Range
variation of mean
standard deviation
Variance
coefficient of variation
standard error of the mean
Measure of central tendency provides a very convenient way of describing a set of scores with a single number that describes the PERFORMANCE of the group.
It is also defined as a single value that is used to describe the “center” of the data.
This document provides an overview of descriptive statistics concepts including measures of central tendency (mean, median, mode), measures of variability (range, standard deviation, variance), and how to compute them from both ungrouped and grouped data. It defines key terms like mean, median, mode, percentiles, quartiles, range, standard deviation, variance, and coefficient of variation. It also discusses how standard deviation can be used to measure financial risk and the empirical rule and Chebyshev's theorem for interpreting standard deviation.
1. The sampling distribution of a statistic is the distribution of all possible values that statistic can take when calculating it from samples of the same size randomly drawn from a population. The sampling distribution will have the same mean as the population but lower variance equal to the population variance divided by the sample size.
2. For a sample mean, the sampling distribution will be approximately normal according to the central limit theorem. A 95% confidence interval for the population mean can be constructed as the sample mean plus or minus 1.96 times the standard error of the mean.
3. For a sample proportion, the sampling distribution will also be approximately normal. A 95% confidence interval can be constructed as the sample proportion plus or minus 1
This document discusses measures of central tendency and variation for numerical data. It defines and provides formulas for the mean, median, mode, range, variance, standard deviation, and coefficient of variation. Quartiles and interquartile range are introduced as measures of spread less influenced by outliers. The relationship between these measures and the shape of a distribution are also covered at a high level.
This document discusses measures of dispersion and the normal distribution. It defines measures of dispersion as ways to quantify the variability in a data set beyond measures of central tendency like mean, median, and mode. The key measures discussed are range, quartile deviation, mean deviation, and standard deviation. It provides formulas and examples for calculating each measure. The document then explains the normal distribution as a theoretical probability distribution important in statistics. It outlines the characteristics of the normal curve and provides examples of using the normal distribution and calculating z-scores.
1. Measures of central tendency include the mean, median, and mode.
2. The mean is the average value found by dividing the sum of all values by the total number of values. The median is the middle value when values are arranged in order. The mode is the value that appears most frequently.
3. For grouped data, the mean is calculated using the sum of the frequency multiplied by the class midpoint divided by the total frequency. The median class is identified which has a cumulative frequency above and below half the total. The mode is the class with the highest frequency.
This document provides information on measures of central tendency and dispersion. It discusses the mean, median, and mode as the three main measures of central tendency. It provides formulas and examples for calculating the mean, median, and mode for both ungrouped and grouped data. The document also covers measures of dispersion including range, semi-interquartile range, variance, standard deviation, and coefficient of variation. It provides formulas and examples for calculating each of these measures. Finally, the document briefly discusses chi-square tests, Pearson's correlation, and using scatterplots to examine relationships between variables.
Unit-I Measures of Dispersion- Biostatistics - Ravinandan A P.pdfRavinandan A P
Biostatistics, Unit-I, Measures of Dispersion, Dispersion
Range
variation of mean
standard deviation
Variance
coefficient of variation
standard error of the mean
Measure of central tendency provides a very convenient way of describing a set of scores with a single number that describes the PERFORMANCE of the group.
It is also defined as a single value that is used to describe the “center” of the data.
This document provides an overview of descriptive statistics concepts including measures of central tendency (mean, median, mode), measures of variability (range, standard deviation, variance), and how to compute them from both ungrouped and grouped data. It defines key terms like mean, median, mode, percentiles, quartiles, range, standard deviation, variance, and coefficient of variation. It also discusses how standard deviation can be used to measure financial risk and the empirical rule and Chebyshev's theorem for interpreting standard deviation.
1. The sampling distribution of a statistic is the distribution of all possible values that statistic can take when calculating it from samples of the same size randomly drawn from a population. The sampling distribution will have the same mean as the population but lower variance equal to the population variance divided by the sample size.
2. For a sample mean, the sampling distribution will be approximately normal according to the central limit theorem. A 95% confidence interval for the population mean can be constructed as the sample mean plus or minus 1.96 times the standard error of the mean.
3. For a sample proportion, the sampling distribution will also be approximately normal. A 95% confidence interval can be constructed as the sample proportion plus or minus 1
The document summarizes key concepts in describing data with numerical measures from a statistics textbook chapter. It covers measures of center including mean, median, and mode. It also covers measures of variability such as range, variance, and standard deviation. It provides examples of calculating these measures and interpreting them, as well as using them to construct box plots.
1. The document discusses various measures of dispersion used to quantify how spread out or variable a data set is. It describes measures such as range, mean deviation, variance, and standard deviation.
2. It also discusses relative measures of dispersion like the coefficient of variation, which allows comparison of variability between data sets with different units or averages. The coefficient of variation expresses variability as a percentage of the mean.
3. Additional concepts covered include skewness, which refers to the asymmetry of a distribution, and kurtosis, which measures the peakedness of a distribution compared to a normal distribution. Positive or negative skewness and leptokurtic, mesokurtic, or platykurtic k
The document discusses various measures of central tendency and variation. It defines mean, median and mode as the three main measures of central tendency. It provides formulas and examples to calculate mean, median and mode for discrete, continuous and grouped data. The document also discusses measures of variation such as range and standard deviation. It provides the formula to calculate standard deviation and an example to demonstrate calculating standard deviation for a set of data.
The document discusses variability and measures of variability. It defines variability as a quantitative measure of how spread out or clustered scores are in a distribution. The standard deviation is introduced as the most commonly used measure of variability, as it takes into account all scores in the distribution and provides the average distance of scores from the mean. Properties of the standard deviation are examined, such as how it does not change when a constant is added to all scores but does change when all scores are multiplied by a constant.
The document discusses various methods for describing data distributions numerically, including measures of center (mean, median), measures of spread (standard deviation, interquartile range), and graphical representations (boxplots). It explains how to calculate and interpret the mean, median, quartiles, five-number summary, standard deviation, and identifies outliers. Choosing an appropriate measure of center and spread depends on the symmetry of the distribution and presence of outliers. Changing the measurement units affects the calculated values but not the underlying shape of the distribution.
This document summarizes various statistical measures used to analyze and describe data distributions, including measures of central tendency (mean, median, mode), dispersion (range, standard deviation, variance), skewness, and kurtosis. It provides formulas and methods for calculating each measure along with interpretations of the results. Measures of central tendency provide a single value to represent the center of the data set. Measures of dispersion describe how spread out or varied the data values are. Skewness and kurtosis measure the symmetry and peakedness of distributions compared to the normal curve.
Unit 1 - Measures of Dispersion - 18MAB303T - PPT - Part 2.pdfAravindS199
The document discusses various measures of dispersion, which describe how data values are spread around the mean. It describes absolute measures like range, interquartile range, mean deviation, and standard deviation. Range is the difference between highest and lowest values. Standard deviation calculates the average distance of all values from the mean. It is the most robust measure as it considers all data points. The document also provides examples of calculating different dispersion measures and their merits and limitations.
This document discusses various methods for summarizing data, including measures of central tendency, dispersion, and categorical data. It describes the mean, median, and mode as measures of central tendency, and how the mean can be affected by outliers while the median is not. Measures of dispersion mentioned include range, standard deviation, variance, and interquartile range. The document also discusses percentiles, standard error, and 95% confidence intervals. Key takeaways are to select appropriate summaries based on the data type and distribution.
This document discusses measures of variability used in statistics. It defines variability as the spread or dispersion of scores. The key measures of variability discussed are the range, variance, and standard deviation. The range is the difference between the highest and lowest scores. The variance is the average of the squared deviations from the mean and represents how far the scores deviate from the mean. The standard deviation is the square root of the variance and represents how much scores typically deviate from the mean. Larger standard deviations indicate greater variability in the scores.
ch-4-measures-of-variability-11 2.ppt for nursingwindri3
This document discusses measures of variability used in statistics. It defines variability as the spread or dispersion of scores. The key measures of variability discussed are the range, variance, and standard deviation. The range is the difference between the highest and lowest scores. The variance is the average of the squared deviations from the mean and represents how far the scores deviate from the mean. The standard deviation is the square root of the variance and represents how much scores typically deviate from the mean. Larger standard deviations indicate greater variability in the scores.
Lecture. Introduction to Statistics (Measures of Dispersion).pptxNabeelAli89
1) The document discusses various measures of dispersion used to quantify how spread out or varied a set of data values are from the average.
2) There are two types of dispersion - absolute dispersion measures how varied data values are in the original units, while relative dispersion compares variability between datasets with different units.
3) Common measures of absolute dispersion include range, variance, and standard deviation. Range is the difference between highest and lowest values, while variance and standard deviation take into account how far all values are from the mean.
The document discusses different measures of central tendency including the mean, median and mode. It provides definitions and formulas for calculating different types of means:
- The arithmetic mean is calculated by summing all values and dividing by the total number of values. It can be calculated using direct or short-cut methods for both individual observations and grouped data.
- Other means include the geometric mean and harmonic mean, which are called special averages.
- The median is the middle value when values are arranged in order. The mode is the value that occurs most frequently.
- Data can be in the form of individual observations, discrete series or continuous series. Formulas are provided for calculating the mean of grouped or ungrouped data
This document discusses measures of skewness in a distribution. It defines skewness as a lack of symmetry such that the mean and median are not equal. There are three main types of distributions: symmetrical, positively skewed, and negatively skewed. The document outlines various absolute and relative measures to quantify skewness, including Karl Pearson's coefficient of skewness, Bowley's coefficient of skewness, and comparing quartiles to the median. Examples are provided to demonstrate calculating these coefficients from data sets.
This document provides an overview of descriptive statistics and numerical summary measures. It discusses measures of central tendency including the mean, median, and mode. It also covers measures of relative standing such as percentiles and quartiles. Additionally, the document outlines measures of dispersion like variance, standard deviation, coefficient of variation, range, and interquartile range. Graphs and charts are presented as ways to describe data using these numerical summary measures.
measure of variability (windri). In research include examplewindri3
This document discusses various measures of variability that can be used to describe how spread out a distribution is. It describes four major measures: range, quartile deviation, average deviation, and standard deviation. The range is the simplest measure, being the difference between the highest and lowest values. The quartile deviation uses the interquartile range to describe the middle 50% of scores. The average deviation takes the average of all deviations from the mean. The standard deviation is the most common measure, being the positive square root of the variance, which is the average of the squared deviations from the mean. Examples are provided for calculating each measure using both grouped and ungrouped data.
The document discusses various statistical concepts including range, mean deviation, variance, and standard deviation. It provides formulas and steps to calculate each measure. The range is the distance between the highest and lowest values. Mean deviation measures the average deviation from the mean. Variance is the average of the squared deviations from the mean and standard deviation is the square root of the variance, representing the average distance from the mean. Examples are given to demonstrate calculating each measure for both ungrouped and grouped data.
This document discusses summary statistics and measures of central tendency and dispersion. It provides examples of how to calculate the mean, mode, median, and geometric mean of data sets. It also discusses how to calculate the standard deviation and variance of data as measures of dispersion. Regression and correlation analysis are introduced as methods to study the relationship between variables and enable prediction. The least squares approach to determining the linear regression line that best fits a data set is demonstrated through an example.
This document provides definitions and explanations of key statistical concepts including:
1. Statistics is defined as the science of collecting, classifying, presenting, and interpreting data. Central tendency measures like mean, median, and mode are used to summarize data.
2. Measures of dispersion like range, interquartile range, mean deviation, and standard deviation describe how spread out the data is from the central tendency. Standard deviation is the most accurate measure as it considers both the deviation from the mean and the mathematical signs.
3. Examples are provided to demonstrate calculating the mean, median, mode, and standard deviation for both ungrouped and grouped data series. The standard deviation provides the best estimation of the population mean when
The document summarizes key concepts in describing data with numerical measures from a statistics textbook chapter. It covers measures of center including mean, median, and mode. It also covers measures of variability such as range, variance, and standard deviation. It provides examples of calculating these measures and interpreting them, as well as using them to construct box plots.
1. The document discusses various measures of dispersion used to quantify how spread out or variable a data set is. It describes measures such as range, mean deviation, variance, and standard deviation.
2. It also discusses relative measures of dispersion like the coefficient of variation, which allows comparison of variability between data sets with different units or averages. The coefficient of variation expresses variability as a percentage of the mean.
3. Additional concepts covered include skewness, which refers to the asymmetry of a distribution, and kurtosis, which measures the peakedness of a distribution compared to a normal distribution. Positive or negative skewness and leptokurtic, mesokurtic, or platykurtic k
The document discusses various measures of central tendency and variation. It defines mean, median and mode as the three main measures of central tendency. It provides formulas and examples to calculate mean, median and mode for discrete, continuous and grouped data. The document also discusses measures of variation such as range and standard deviation. It provides the formula to calculate standard deviation and an example to demonstrate calculating standard deviation for a set of data.
The document discusses variability and measures of variability. It defines variability as a quantitative measure of how spread out or clustered scores are in a distribution. The standard deviation is introduced as the most commonly used measure of variability, as it takes into account all scores in the distribution and provides the average distance of scores from the mean. Properties of the standard deviation are examined, such as how it does not change when a constant is added to all scores but does change when all scores are multiplied by a constant.
The document discusses various methods for describing data distributions numerically, including measures of center (mean, median), measures of spread (standard deviation, interquartile range), and graphical representations (boxplots). It explains how to calculate and interpret the mean, median, quartiles, five-number summary, standard deviation, and identifies outliers. Choosing an appropriate measure of center and spread depends on the symmetry of the distribution and presence of outliers. Changing the measurement units affects the calculated values but not the underlying shape of the distribution.
This document summarizes various statistical measures used to analyze and describe data distributions, including measures of central tendency (mean, median, mode), dispersion (range, standard deviation, variance), skewness, and kurtosis. It provides formulas and methods for calculating each measure along with interpretations of the results. Measures of central tendency provide a single value to represent the center of the data set. Measures of dispersion describe how spread out or varied the data values are. Skewness and kurtosis measure the symmetry and peakedness of distributions compared to the normal curve.
Unit 1 - Measures of Dispersion - 18MAB303T - PPT - Part 2.pdfAravindS199
The document discusses various measures of dispersion, which describe how data values are spread around the mean. It describes absolute measures like range, interquartile range, mean deviation, and standard deviation. Range is the difference between highest and lowest values. Standard deviation calculates the average distance of all values from the mean. It is the most robust measure as it considers all data points. The document also provides examples of calculating different dispersion measures and their merits and limitations.
This document discusses various methods for summarizing data, including measures of central tendency, dispersion, and categorical data. It describes the mean, median, and mode as measures of central tendency, and how the mean can be affected by outliers while the median is not. Measures of dispersion mentioned include range, standard deviation, variance, and interquartile range. The document also discusses percentiles, standard error, and 95% confidence intervals. Key takeaways are to select appropriate summaries based on the data type and distribution.
This document discusses measures of variability used in statistics. It defines variability as the spread or dispersion of scores. The key measures of variability discussed are the range, variance, and standard deviation. The range is the difference between the highest and lowest scores. The variance is the average of the squared deviations from the mean and represents how far the scores deviate from the mean. The standard deviation is the square root of the variance and represents how much scores typically deviate from the mean. Larger standard deviations indicate greater variability in the scores.
ch-4-measures-of-variability-11 2.ppt for nursingwindri3
This document discusses measures of variability used in statistics. It defines variability as the spread or dispersion of scores. The key measures of variability discussed are the range, variance, and standard deviation. The range is the difference between the highest and lowest scores. The variance is the average of the squared deviations from the mean and represents how far the scores deviate from the mean. The standard deviation is the square root of the variance and represents how much scores typically deviate from the mean. Larger standard deviations indicate greater variability in the scores.
Lecture. Introduction to Statistics (Measures of Dispersion).pptxNabeelAli89
1) The document discusses various measures of dispersion used to quantify how spread out or varied a set of data values are from the average.
2) There are two types of dispersion - absolute dispersion measures how varied data values are in the original units, while relative dispersion compares variability between datasets with different units.
3) Common measures of absolute dispersion include range, variance, and standard deviation. Range is the difference between highest and lowest values, while variance and standard deviation take into account how far all values are from the mean.
The document discusses different measures of central tendency including the mean, median and mode. It provides definitions and formulas for calculating different types of means:
- The arithmetic mean is calculated by summing all values and dividing by the total number of values. It can be calculated using direct or short-cut methods for both individual observations and grouped data.
- Other means include the geometric mean and harmonic mean, which are called special averages.
- The median is the middle value when values are arranged in order. The mode is the value that occurs most frequently.
- Data can be in the form of individual observations, discrete series or continuous series. Formulas are provided for calculating the mean of grouped or ungrouped data
This document discusses measures of skewness in a distribution. It defines skewness as a lack of symmetry such that the mean and median are not equal. There are three main types of distributions: symmetrical, positively skewed, and negatively skewed. The document outlines various absolute and relative measures to quantify skewness, including Karl Pearson's coefficient of skewness, Bowley's coefficient of skewness, and comparing quartiles to the median. Examples are provided to demonstrate calculating these coefficients from data sets.
This document provides an overview of descriptive statistics and numerical summary measures. It discusses measures of central tendency including the mean, median, and mode. It also covers measures of relative standing such as percentiles and quartiles. Additionally, the document outlines measures of dispersion like variance, standard deviation, coefficient of variation, range, and interquartile range. Graphs and charts are presented as ways to describe data using these numerical summary measures.
measure of variability (windri). In research include examplewindri3
This document discusses various measures of variability that can be used to describe how spread out a distribution is. It describes four major measures: range, quartile deviation, average deviation, and standard deviation. The range is the simplest measure, being the difference between the highest and lowest values. The quartile deviation uses the interquartile range to describe the middle 50% of scores. The average deviation takes the average of all deviations from the mean. The standard deviation is the most common measure, being the positive square root of the variance, which is the average of the squared deviations from the mean. Examples are provided for calculating each measure using both grouped and ungrouped data.
The document discusses various statistical concepts including range, mean deviation, variance, and standard deviation. It provides formulas and steps to calculate each measure. The range is the distance between the highest and lowest values. Mean deviation measures the average deviation from the mean. Variance is the average of the squared deviations from the mean and standard deviation is the square root of the variance, representing the average distance from the mean. Examples are given to demonstrate calculating each measure for both ungrouped and grouped data.
This document discusses summary statistics and measures of central tendency and dispersion. It provides examples of how to calculate the mean, mode, median, and geometric mean of data sets. It also discusses how to calculate the standard deviation and variance of data as measures of dispersion. Regression and correlation analysis are introduced as methods to study the relationship between variables and enable prediction. The least squares approach to determining the linear regression line that best fits a data set is demonstrated through an example.
This document provides definitions and explanations of key statistical concepts including:
1. Statistics is defined as the science of collecting, classifying, presenting, and interpreting data. Central tendency measures like mean, median, and mode are used to summarize data.
2. Measures of dispersion like range, interquartile range, mean deviation, and standard deviation describe how spread out the data is from the central tendency. Standard deviation is the most accurate measure as it considers both the deviation from the mean and the mathematical signs.
3. Examples are provided to demonstrate calculating the mean, median, mode, and standard deviation for both ungrouped and grouped data series. The standard deviation provides the best estimation of the population mean when
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2. • Measures of central tendency of a distribution -
A numerical value that describes the central position of data
• 3 common measures
• Mean
• Median
• Mode
3. Arithmetic mean:
It is obtained by summing up of all
observations divided by number of
observations.
It is denoted by X (Sample Mean)
Population mean is Denoted by µ (mu)
4. Mean (Arithmetic Mean)
Mean (arithmetic mean) of data values
Sample mean
Population mean
1 1 2
n
i
i n
X
X X X
X
n n
1 1 2
N
i
i N
X
X X X
N N
Sample Size
Size Population
5. • Measures of Location: Averages (Mean)
Mean = Total or sum of the observations
Number of observations
= X1 + X2 + …. Xn = ΣX
n n
• The mean is calculated by different methods in
two types of series, ungrouped and grouped
series.
17/06/2024 Measures of Central tendancy 5
6. Ungrouped Series:
In such series the number of observations is small and there are two methods for
calculating the mean. The choice depends upon the size of observations in the series.
(I). When the observations are small in size, simply add them up and divide by the
number of observations.
Example: 1. Tuberculin test reaction of 10 boys is arranged in ascending order being
measured in millimeters. Find the mean size of reaction: 3,5,7,7,8,8,9,10,11,12
→ Mean or = ΣX
N
= 3+5+7+7+8+8+9+10+11+12
10
= 80
10
=8 mm
7. Examples 2. Height in Centimeters for 7 school children are given below.
• 148, 143, 160, 152,157, 150, 155 Cms.
• Find the mean.
• By direct method
= ΣX = 1065 = 152.1 Centimeters
n 7
17 June 2024 7
Mean, Median and Mode
8. By assumed mean (w) method
X X-w=x (w=140) x
148 148-140 8
143 143-140 3
160 160-140 20
152 152-140 12
157 157-140 17
150 150-140 10
155 155-140 15
Σx= 85
x = Σ(X-w) = 85 =12.1
n 7
= w + x
= 140 + 12.1
= 152.1Cm
17 June 2024 8
Mean, Median and Mode
9. Example. The average income of 10 lady doctors is Rs. 25000/- per month and that
of 20 male doctors is Rs. 35000/- per month. Calculate the weighted mean or
average income of all doctors.
→
For lady doctors X1 = Rs. 25000/- f1= 10
For male doctors X2 = Rs. 35000/- f2= 20
n = f1 + f2 = 10+20 =30
Total Income of lady doctors = X1 x f1 = ΣfX1 = 25000 x 10= 2,50,000
Total income of male doctors = X2 x f2 = ΣfX2 = 35000 x 20 =7,00,000
Total income of all doctors are= ΣfX = ΣfX1 + ΣfX2 =2,50,000 + 7,00,000=
9,50,000
The weighted mean income of all the doctors = ΣfX = 9,50,000 = Rs. 31,666.66
n 30
So, average income of all doctors is Rs. 31,666.66
Measures of Central tendancy 9
10. Example: Find the average weight of college students in kilogram from
the table given below.
Weight of students in Kg No. of students
60-<61 10
61- 20
62- 45
63- 50
64- 60
65- 40
66-<67 15
Total 240
17 June 2024 10
Mean, Median and Mode
11. 64- 60
65- 40
66-<67 15
Total 240
→
1st
Method:
Weight of
students in Kg
X
Mid-point of
each group
Xg
No. of
students
f fXg
60-<61 60.5 10 605
61- 61.5 20 1230
62- 62.5 45 2812.5
63- 63.5 50 3175
64- 64.5 60 3870
65- 65.5 40 2620
66-<67 66.5 15 997.5
Total n=240 ΣfXg=15310
Now, = ΣfXg = 15310 = 63.79Kg
n 240
So, Mean weight of college students is 63.79Kg
17/06/2024 Measures of Central tendancy 11
12. 17/06/2024 Measures of Central tendancy 12
Weight of
students in
Kg X
Mid-point of
each group
Xg
No. of
students
f
Working
units
x
Groups
weight
fx
Sum of fx
60-<61 60.5 10 -2 -20
61- 61.5 20 -1 -20
62- 62.5 (w) 45 0 0 -40
63- 63.5 50 +1 50
64- 64.5 60 +2 120
65- 65.5 40 +3 120
66-<67 66.5 15 +4 60 +350
Total n=240 Σfx=+310
13. 64- 64.5 60 +2 120
65- 65.5 40 +3 120
66-<67 66.5 15 +4 60 +350
Total n=240 Σfx=+310
Mean in working units
= Σfx = 310 =1.29
n 240
Mean in real units
= w + x Group interval
= 62.5 + 1.29
= 63.79 Kg
So, mean weight of college students is 63.79Kg
17/06/2024 Measures of Central tendancy 13
14. MEDIAN
It is the value of middle observation after
placing the observations in either ascending or
descending order.
Half the values lie above it and half below it.
15. UNGROUPED SERIES
• If the number of observations is
odd then median of the data will be
n+1/2th observation
• If even then median of the data will be
the average of n/2th and ( n/2 ) +1th
16. Example 1: To find the median of 4,5,7,2,1 [ODD].
Step 1: Count the total numbers given.
There are 5 elements or numbers in the
distribution.
Step 2: Arrange the numbers in ascending order.
1,2,4,5,7
Step 3: The total elements in the distribution (5) is
odd.
The middle position can be calculated using the
formula. (n+1)/2
So the middle position is (5+1)/2 = 6/2 = 3 th Value
The number at 3rd position is = Median = 4
17. Example 2 : To find the median of 5,7,2,1,6,4.
step 1 : count the total numbers given.
there are 6 numbers in the distribution.
step 2 :arrange the numbers in ascending
order.
1,2,4,5,6,7.
step 3 :the total numbers in the distribution is 6
(even).
so the average of two numbers which are
respectively in positions n/2th and (n/2)+1th will
be the median of the given data.
Median = (4+5)/2 = 4.5
18. Mode
• A measure of central tendency
• Value that occurs most often
• Not affected by extreme values
• There may be no mode or several modes
19. • To find the mode of 11,3,5,11,7,3,11
• Arrange the numbers in ascending order.
3,3,5,7,11,11,11
Mode = 11
20. Measures of variability of individual
observations:
• i. Range
• ii. Interquartile range
• iii. Mean deviation
• iv. Standard deviation
• v. Coefficient of variation.
21. Measures of variability of samples:
• i. Standard error of mean
• ii Standard error of difference between two means
• iii Standard error of proportion
• iv Standard error of difference between two proportions
• v. Standard error of correlation coefficient
• vi. Standard deviation of regression coefficient.
22. 22
The Range
• The range is defined as the difference between the largest
score in the set of data and the smallest score in the set of
data,
• XL - XS
• What is the range of the following data:
4 8 1 6 6 2 9 3 6 9
• The largest score (XL) is 9; the smallest score (XS) is 1; the range
is XL - XS = 9 - 1 = 8
23. Quartiles
•Split Ordered Data into 4 Quarters
•
• Position of i-th Quartile: position of point
25% 25% 25% 25%
Q1 Q2
Q3
Q i(n+1)
i 4
Data in Ordered Array: 11 12 13 16 16 17 18 21 22
Position of Q1 = 2.50 Q1 =12.5
= 1•(9 + 1)
4
24. • Measure of Variation
• Also Known as Midspread:
Spread in the Middle 50%
• Difference Between Third & First
Quartiles: Interquartile Range =
• Not Affected by Extreme Values
Interquartile Range
1
3 Q
Q
Data in Ordered Array: 11 12 13 16 16 17 18 21 22
1
3 Q
Q = 17.5 - 12.5 = 5
27. • 𝑋 =
𝑋
𝑛
=
775
5
= 155 cm
• Mean deviation MD =
X−𝑋
𝑛
=
40
5
= 8 cm
• Mean deviation is not used in statistical analysis being less
mathematical value, particularly in drawing inferences.
Observations (X) X − 𝑋 X − 𝑋
150 -5 5
160 +5 5
155 0 0
170 +15 15
140 -15 15
ΣX= 775 Σ X − 𝑋 = 40
e.g. Height of 5 students in Centimeter
28. • Root-mean squared deviation called SD.
SD =
X−𝑋 2
𝑛−1
When sample size is less than 30
• The formula becomes
• SD =
X−𝑋 2
𝑛
• When sample size is more than 30
29. Observation
X
Deviation from Mean
x= X-
Square of deviation
x2 =( X - )2
23 +3 9
22 +2 4
20 0 0
24 +4 16
16 -4 16
17 -3 9
18 -2 4
19 -1 1
21 +1 1
ΣX=180 0 Σ=( X - )2=60
Calculation of Standard Deviations in Ungrouped series:
Example:
Find the mean respiratory rate per minute and its SD when in 9 cases the rate
was found to be 23, 22, 20, 24, 16, 17, 18, 19 and 21.
= ΣX = 180 = 20/minute
N 9
So s or SD=
= =2.74 min
Σ(X - )2
n -1
60/9-1
31. • It is a measure used to compare relative variability
5. Coefficientof Variation:
32. Persons Mean Ht in Cm SD in Cm
Adults 160cm 10cm
Children 60cm 5cm
In two series of adults aged 21 years and children 3 months old following values were
obtained for height. Find which series shows greater variation?
CV = SD x 100
Mean
CV of adults = 10 x 100 = 6.25%
160
CV of children = 5 x 100 = 8.33%
60
Thus it is found that heights in children show greater variation than in
adults.
35. (a) The area between one standard deviation( SD) on either side of the
mean ( x ± l ϭ ) will include approximately 68% of the values in
the distribution
(b) The area between two standard deviations on either side of the
mean ( x ± 2 ϭ ) will cover most of the values, i.e., approximately 95
% of the values
(c) The area between three standard deviations on either side of the
mean ( x ± 3 ϭ) will include 99.7 % of the values.
• These limits on either side of the mean are called “confidence
limits
36. Properties of
STANDARD NORMAL CURVE
• Bell shaped & smooth curve
• Two tailed & symmetrical
• Tail doesn’t touch the base line
• Area under the curve is 1
• Mean = 0
• Standard deviation is =1
• Mean, Median, Mode coincide
• Two inflection- Convex at centre, convert to Concave while descending to periphery
• Perpendicular drawn from the point of inflection cut the base at 1 standard deviation
• Approximately 68%, 95%, 99% observations are included in the range of Mean +/- 1SD, 2 SD, 3 SD
respectively
• No portion of the curve lie below the X axis
-5 -4 -3 -2 -1 0 1 2 3 4 5
68.26%
95.44%
99.72%
37. Example
Q: 95% of students at school are between 1.1m and 1.7m tall.
Assuming this data is normally distributed calculate the mean
and standard deviation?
38. 95% is 2 standard deviations either side of the mean (a total of
4 standard deviations) so:
1 standard deviation = (1.7m-1.1m) / 4
= 0.6m / 4 = 0.15m
And this is the result:
The mean is halfway between 1.1m and 1.7m:
Mean = (1.1m + 1.7m) / 2
= 1.4m
39. Q: 68% of the marks in a test are between 51 and 64, Assuming this
data is normally distributed, what are the mean and standard
deviation?
Example
40. Answer:
The mean is halfway between 51 and 64:
Mean = (51 + 64)/2 = 57.5
68% is 1 standard deviation either side of the
mean (a total of 2 standard deviations)
so:
1 standard deviation = (64 - 51)/2 = 13/2 = 6.5
41. Example
Q: Average weight of baby at birth is 3.05 kg with SD of 0.39kg. If the
birth weights are normally distributed would you regard:
1.Weight of 4 kg as abnormal?
2. Weight of 2.5 kg as normal?
42. • Answer
Normal limits of weight will be within range of
Mean + 2 SD
= 3.05 + (2 x 0.39)
= 3.05 + 0.78
= 2.27 to 3.83
1. The wt of 4 kg lies outside the normal limits. So it is taken as
abnormal.
2. The wt. of 2.5 kg lies within the normal limits. So it is taken as
normal.
43. Standard normal deviate
The distance of a value (x) from the mean (X bar) of the
curve in units of standard deviation is called “relative
deviate or standard normal deviate” and usually denoted
by Z.
Z = Observation –Mean
Standard Deviation
44. Q: The pulse of a group of normal healthy
males was 72, with a standard deviation of 2.
What is the probability that a male chosen at
random would be found to have a pulse of 80
or more ?
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