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.
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.
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.
Lecture 3 Measures of Central Tendency and Dispersion.pptxshakirRahman10
Objectives:
Define measures of central tendency (mean, median, and mode)
Define measures of dispersion (variance and standard deviation).
Compute the measures of central tendency and Dispersion.
Learn the application of mean and standard deviation using Empirical rule and Tchebyshev’s theorem.
Measures of Central Tendency:
A measure of the central tendency is a value about which the observations tend to cluster.
In other words it is a value around which a data set is centered.
The three most common measures of central tendency are mean, median and mode.
A measure of the central tendency is a value about which the observations tend to cluster.
In other words it is a value around which a data set is centered.
The three most common measures of central tendency are mean, median and mode.
A measure of the central tendency is a value about which the observations tend to cluster.
In other words it is a value around which a data set is centered.
The three most common measures of central tendency are mean, median and mode.
A measure of the central tendency is a value about which the observations tend to cluster.
In other words it is a value around which a data set is centered.
The three most common measures of central tendency are mean, median and mode.
Why is it needed?
To summarize the data.
It provides with a typical value that gives the picture of the entire data set
Mean:
It is the arithmetic average of a set of numbers, It is the most common measure of central tendency.
Computed by summing all values in the data set and dividing the sum by the number of values in the data set Properties:
Applicable for interval and ratio data
Not applicable for nominal or ordinal data
Affected by each value in the data set, including extreme values.
Formula:
Mean is calculated by adding all values in the data set and dividing the sum by the number of values in the data set.
Median:
Mid-point or Middle value of the data when the measurements are arranged in ascending order.
A point that divides the data into two equal parts.
Computational Procedure:
Arrange the observations in an ascending order.
If there is an odd number of terms, the median is the middle value and If there is an even number of terms, the median is the average of the middle two terms.
Mode:
The mode is the observation that occurs most frequently in the data set.
There can be more than one mode for a data set OR there maybe no mode in a data set.
Is also applicable to the nominal data.
Comparison of Measures of Central Tendency in Positively Skewed Distributions:
Majority of the data values fall to the left of the mean and cluster at the lower end of the distribution: the tail is to the right Mean, median & mode are different When a distribution has a few extremely high scores, the mean will have a greater value than the median = positively skewed.
Majority of the data values fall to
the right of the mean and cluster at the upper end of the distribution= Negatively Skewed
Lecture 3 Measures of Central Tendency and Dispersion.pptxshakirRahman10
Objectives:
Define measures of central tendency (mean, median, and mode)
Define measures of dispersion (variance and standard deviation).
Compute the measures of central tendency and Dispersion.
Learn the application of mean and standard deviation using Empirical rule and Tchebyshev’s theorem.
Measures of Central Tendency:
A measure of the central tendency is a value about which the observations tend to cluster.
In other words it is a value around which a data set is centered.
The three most common measures of central tendency are mean, median and mode.
A measure of the central tendency is a value about which the observations tend to cluster.
In other words it is a value around which a data set is centered.
The three most common measures of central tendency are mean, median and mode.
A measure of the central tendency is a value about which the observations tend to cluster.
In other words it is a value around which a data set is centered.
The three most common measures of central tendency are mean, median and mode.
A measure of the central tendency is a value about which the observations tend to cluster.
In other words it is a value around which a data set is centered.
The three most common measures of central tendency are mean, median and mode.
Why is it needed?
To summarize the data.
It provides with a typical value that gives the picture of the entire data set
Mean:
It is the arithmetic average of a set of numbers, It is the most common measure of central tendency.
Computed by summing all values in the data set and dividing the sum by the number of values in the data set Properties:
Applicable for interval and ratio data
Not applicable for nominal or ordinal data
Affected by each value in the data set, including extreme values.
Formula:
Mean is calculated by adding all values in the data set and dividing the sum by the number of values in the data set.
Median:
Mid-point or Middle value of the data when the measurements are arranged in ascending order.
A point that divides the data into two equal parts.
Computational Procedure:
Arrange the observations in an ascending order.
If there is an odd number of terms, the median is the middle value and If there is an even number of terms, the median is the average of the middle two terms.
Mode:
The mode is the observation that occurs most frequently in the data set.
There can be more than one mode for a data set OR there maybe no mode in a data set.
Is also applicable to the nominal data.
Comparison of Measures of Central Tendency in Positively Skewed Distributions:
Majority of the data values fall to the left of the mean and cluster at the lower end of the distribution: the tail is to the right Mean, median & mode are different When a distribution has a few extremely high scores, the mean will have a greater value than the median = positively skewed.
Majority of the data values fall to
the right of the mean and cluster at the upper end of the distribution= Negatively Skewed
We understand the unique challenges pickleball players face and are committed to helping you stay healthy and active. In this presentation, we’ll explore the three most common pickleball injuries and provide strategies for prevention and treatment.
CHAPTER 1 SEMESTER V - ROLE OF PEADIATRIC NURSE.pdfSachin Sharma
Pediatric nurses play a vital role in the health and well-being of children. Their responsibilities are wide-ranging, and their objectives can be categorized into several key areas:
1. Direct Patient Care:
Objective: Provide comprehensive and compassionate care to infants, children, and adolescents in various healthcare settings (hospitals, clinics, etc.).
This includes tasks like:
Monitoring vital signs and physical condition.
Administering medications and treatments.
Performing procedures as directed by doctors.
Assisting with daily living activities (bathing, feeding).
Providing emotional support and pain management.
2. Health Promotion and Education:
Objective: Promote healthy behaviors and educate children, families, and communities about preventive healthcare.
This includes tasks like:
Administering vaccinations.
Providing education on nutrition, hygiene, and development.
Offering breastfeeding and childbirth support.
Counseling families on safety and injury prevention.
3. Collaboration and Advocacy:
Objective: Collaborate effectively with doctors, social workers, therapists, and other healthcare professionals to ensure coordinated care for children.
Objective: Advocate for the rights and best interests of their patients, especially when children cannot speak for themselves.
This includes tasks like:
Communicating effectively with healthcare teams.
Identifying and addressing potential risks to child welfare.
Educating families about their child's condition and treatment options.
4. Professional Development and Research:
Objective: Stay up-to-date on the latest advancements in pediatric healthcare through continuing education and research.
Objective: Contribute to improving the quality of care for children by participating in research initiatives.
This includes tasks like:
Attending workshops and conferences on pediatric nursing.
Participating in clinical trials related to child health.
Implementing evidence-based practices into their daily routines.
By fulfilling these objectives, pediatric nurses play a crucial role in ensuring the optimal health and well-being of children throughout all stages of their development.
CRISPR-Cas9, a revolutionary gene-editing tool, holds immense potential to reshape medicine, agriculture, and our understanding of life. But like any powerful tool, it comes with ethical considerations.
Unveiling CRISPR: This naturally occurring bacterial defense system (crRNA & Cas9 protein) fights viruses. Scientists repurposed it for precise gene editing (correction, deletion, insertion) by targeting specific DNA sequences.
The Promise: CRISPR offers exciting possibilities:
Gene Therapy: Correcting genetic diseases like cystic fibrosis.
Agriculture: Engineering crops resistant to pests and harsh environments.
Research: Studying gene function to unlock new knowledge.
The Peril: Ethical concerns demand attention:
Off-target Effects: Unintended DNA edits can have unforeseen consequences.
Eugenics: Misusing CRISPR for designer babies raises social and ethical questions.
Equity: High costs could limit access to this potentially life-saving technology.
The Path Forward: Responsible development is crucial:
International Collaboration: Clear guidelines are needed for research and human trials.
Public Education: Open discussions ensure informed decisions about CRISPR.
Prioritize Safety and Ethics: Safety and ethical principles must be paramount.
CRISPR offers a powerful tool for a better future, but responsible development and addressing ethical concerns are essential. By prioritizing safety, fostering open dialogue, and ensuring equitable access, we can harness CRISPR's power for the benefit of all. (2998 characters)
R3 Stem Cells and Kidney Repair A New Horizon in Nephrology.pptxR3 Stem Cell
R3 Stem Cells and Kidney Repair: A New Horizon in Nephrology" explores groundbreaking advancements in the use of R3 stem cells for kidney disease treatment. This insightful piece delves into the potential of these cells to regenerate damaged kidney tissue, offering new hope for patients and reshaping the future of nephrology.
CHAPTER 1 SEMESTER V PREVENTIVE-PEDIATRICS.pdfSachin Sharma
This content provides an overview of preventive pediatrics. It defines preventive pediatrics as preventing disease and promoting children's physical, mental, and social well-being to achieve positive health. It discusses antenatal, postnatal, and social preventive pediatrics. It also covers various child health programs like immunization, breastfeeding, ICDS, and the roles of organizations like WHO, UNICEF, and nurses in preventive pediatrics.
Navigating Challenges: Mental Health, Legislation, and the Prison System in B...Guillermo Rivera
This conference will delve into the intricate intersections between mental health, legal frameworks, and the prison system in Bolivia. It aims to provide a comprehensive overview of the current challenges faced by mental health professionals working within the legislative and correctional landscapes. Topics of discussion will include the prevalence and impact of mental health issues among the incarcerated population, the effectiveness of existing mental health policies and legislation, and potential reforms to enhance the mental health support system within prisons.
QA Paediatric dentistry department, Hospital Melaka 2020Azreen Aj
QA study - To improve the 6th monthly recall rate post-comprehensive dental treatment under general anaesthesia in paediatric dentistry department, Hospital Melaka
How many patients does case series should have In comparison to case reports.pdfpubrica101
Pubrica’s team of researchers and writers create scientific and medical research articles, which may be important resources for authors and practitioners. Pubrica medical writers assist you in creating and revising the introduction by alerting the reader to gaps in the chosen study subject. Our professionals understand the order in which the hypothesis topic is followed by the broad subject, the issue, and the backdrop.
https://pubrica.com/academy/case-study-or-series/how-many-patients-does-case-series-should-have-in-comparison-to-case-reports/
2. Definition:
• Constructing frequency distribution of raw data is the first step towards
condensation of large data into compact form.
• It is necessary to condense the data into a single value. Such a single value is
called an average.
3. Definition:
• In most of the data the average is a centre of concentration of the values in
the data therefore, the average is called a measure of central tendency.
• The central tendency is stated as the statistical measure that represents the
single value of the entire distribution or a dataset. It aims to provide an
accurate description of the entire data in the distribution.
4. Properties of a Central Tendency:
• It should be rigidly defined
• Its computation should be based on all observations
• It should lend itself for algebraic treatment
• It should be least affected by extreme observations.
5. The following are the different measures of central
tendency:
1. Average ( Arithmetic mean)
2. Median
3. Mode
4. Quartiles
5. Geometric Mean
6. Harmonic Mean
7. Weighted Mean
6. Average/ Arithmetic Mean(AM):
• This is commonly used. Arithmetic mean (AM) or mean is a sum of all
observations divided by number of observations.
Computation for Ungrouped data:
The mean of n observations X1, X2……..Xn is given by
A.M = X1 + X2 + ……. +Xn
n
= Sum of observations/ Number of observations
7. Notation form :
• 𝐴𝑀 =
𝑥
𝑛
=
𝑠𝑢𝑚𝑥
𝑛
• Arithmetic Mean is denoted by 𝑥. The notation ∑ is read as sigma and 𝑥 as X
bar.
8. Merits Of AM:
• Merits:
1. It is easy to calculate and understand
2. It is based on all observations
3. It is familiar to common man and rigidly defined
4. It is capable of further mathematical treatment
5. It is least affected by sampling fluctuations. Hence it is more stable.
9. Demerits Of AM
• It is used only for quantitative data.
• It is unduly affected by extreme observations
• It cannot be calculated when the frequency distribution is with open end
classes.
• It cannot be determined graphically
• Sometimes AM may not be an observation in a data
10. Example:
Q. Obtain the arithmetic mean of marks scored by a student in 8 unit tests of II
MBBS Class.
58 62 67 65 68 70 69 61
12. Short cut or Assumed mean method:
• When observations in data set are large in size, it is a laborious work to find
mean. To avoid this difficulty, short cut method is adopted.
• Assume arbitrary mean i.e., an value from data set (which will simplify the
calculations) and subtract this assumed mean from each observation.
• We get what is known as differences or deviations.
• Obtain mean for deviations by usual method.
13. Contd….
• Observations
• Original data: X1, X2, ………Xn
• Differences or X1-a, X2-a, …….. Xn-a
• Deviations: d1, d2,…..dn
• Where a is any value from dataset.
• Mean for deviations(d) = sum d/n. Thus, Mean of original data(X)=a+d
14. Example:
• In a series of 10 postmorterms following observations regarding weight (in
gms) of liver were found.
• 1420 1405 1425 1410 1415
1435 1430 1415 1445 1430
16. Computation of grouped data
• In Statistics, data plays a vital role in estimating the different types of parameters. To
draw any conclusions from the given data, first, we need to arrange the data in such a
way that one can perform suitable statistical experiments. We know that data can be
grouped into two ways, namely, Discrete and Continuous frequency distribution.
17. Discrete frequency distribution:
• Suppose we have X1, X2, …….. Xn observations with corresponding
frequencies f1, f2,…..,fn. The AM is defined as
• 𝑥 =
𝑓1𝑥1+𝑓2𝑥2+⋯+𝑓𝑛𝑥𝑛
𝑓1+𝑓2
+…+𝑓𝑛
• In notation form, we have
• MeanX= ∑(f.x)/ ∑f
= ∑(f.x)/N
= Sum (Frequency×observation)
• Total Frequency
18. Calculate the average number of children per
family from the following data:
NO: of children No: of families
0 30
1 52
2 60
3 65
4 18
5 10
6 05
19. Solution:
NO: of Children
(X)
NO: of families
(f)
Total NO: of Children
(f.x)
0 30 0×30=0
1 52 1×52=52
2 60 2×60=120
3 65 3×65=195
4 18 4×18=72
5 10 5×10=50
6 5 6×5=30
Total 240 519
21. Continuous frequency distribution:
• In continuous frequency distribution, the frequency is not associated with
any specified single value but spread over entire class.
• It creates difficulty for finding mid values X1, X2,….,Xn. To overcome this
difficulty, we make a reasonable assumption that the frequency is associated
with mid-value of class, or the frequency is distributed uniformly over the
entire class.
• Mean (X) = Sum(f.x)/ Sum(f)
22. The following are different steps to calculate average
for continuous frequency distribution
• Step 1- Write all class intervals serially in the first column and corresponding
frequency in the second column.
• Step 2- The mid values of each class interval are obtained by adding lower
and upper class interval and dividing resultant quantity by 2 and put these
values in third column.
• Step 3- Multiply each ‘f’ by corresponding X and write this product in fourth
column. The addition of this column gives sum(fx). i.e ∑f.x.
23. Notation form:
• X= Sum of fourth column
Sum of second column
= Sum (f.x)
Sum(f)
24. Example:
• Find the average age (in years) at the time of death in city A.
Age Interval NO: of Deaths
0-10 16
10-20 09
20-30 20
30-40 11
40-50 07
50-60 12
60-70 09
70-80 04
80-90 02
27. 2. MEDIAN
• The mean is unduly affected by extreme observations and cannot be
calculated for distribution with open end class and qualitative variables like
honesty, sex, religion etc.
• To overcome these drawbacks, we use other measures of central tendency
like median.
28. Definition:
• When all the observations of a variable are arranged in either ascending or
descending order, the middle observation is known as median. It divides the
whole data into two equal portions.
• In other words, 50% of the observations will be smaller than the median
while 50% of the observations will be larger than it.
29. Merits:
• It is easy to understand and easy to calculate
• It can be computed for a distribution with open end classes.
• It is not affected due to extreme observations
• It is applicable for quantitative as well as qualitative data.
• It can be determined graphically.
30. Demerits:
• It is not based on all the observations, hence it is not proper representative.
• It is not as rigidly defined as the arithmetic mean.
• It is not capable of further mathematical treatment.
31. Computation of Median:
Ungrouped Data:
• As discussed above, the median is one of the measures of central tendency,
which gives the middle value of the given data set.
• While finding the median of the ungrouped data, first arrange the given data
in ascending order, and then find the median value.
32. • If the total number of observations (n) is odd, then the median is (n+1)/2 th
observation.
• If the total number of observations (n) is even, then the median will be average of
n/2th and the (n/2)+1 th observation.
33. Example:
For example, 6, 4, 7, 3 and 2 is the given data set.
• To find the median of the given dataset, arrange it in ascending order.
• Therefore, the dataset is 2, 3, 4, 6 and 7.
• In this case, the number of observations is odd. (i.e) n= 5
• Hence, median = (n+1)/2 th observation.
• Median = (5+1)/2 = 6/2 = 3rd observation.
• Therefore, the median of the given dataset is 4
34. Calculation for grouped data
• In a grouped data, it is not possible to find the median for the given observation by
looking at the cumulative frequencies. The middle value of the given data will be in
some class interval. So, it is necessary to find the value inside the class interval that
divides the whole distribution into two halves.
• we have to find the median class.
• To find the median class, we have to find the cumulative frequencies of all the classes
and n/2. After that, locate the class whose cumulative frequency is greater than (nearest
to) n/2. The class is called the median class.
37. Solution:
• To find the median height, first, we need to find the class intervals and their corresponding frequencies.
• The given distribution is in the form of being less than type,145, 150 …and 165 gives the upper limit. Thus,
the class should be below 140, 140-145, 145-150, 150-155, 155-160 and 160-165.
• From the given distribution, it is observed that,
• 4 girls are below 140. Therefore, the frequency of class intervals below 140 is 4.
• 11 girls are there with heights less than 145, and 4 girls with height less than 140
• Hence, the frequency distribution for the class interval 140-145 = 11-4 = 7
• Likewise, the frequency of 145 -150= 29 – 11 = 18
• Frequency of 150-155 = 40-29 = 11
• Frequency of 155 – 160 = 46-40 = 6
• Frequency of 160-165 = 51-46 = 5
38. Therefore, the frequency distribution table along
with the cumulative frequencies are given below:
Class Intervals Frequency Cumulative Frequency
Below 140 4 4
140 – 145 7 11
145 – 150 18 29
150 – 155 11 40
155 – 160 6 46
160 – 165 5 51
39. Contd….
• Here, n= 51.
• Therefore, n/2 = 51/2 = 25.5
• Thus, the observations lie between the class interval 145-150, which is called the
median class.
• Therefore,
• Lower class limit = 145
• Class size, h = 5
• Frequency of the median class, f = 18
• Cumulative frequency of the class preceding the median class, cf = 11.
40. • Now, substituting the values in the formula, we get
• Median=145+(25.5−11/18)×5
• Median = 145 + (72.5/18)
• Median = 145 + 4.03
• Median = 149.03.
• Therefore, the median height for the given data is 149. 03 cm.
42. MODE:
• In statistics, the mode is the value that is repeatedly occurring in a given set.
We can also say that the value or number in a data set, which has a high
frequency or appears more frequently, is called mode or modal value. It is
one of the three measures of central tendency, apart from mean and median.
For example, the mode of the set {3, 7, 8, 8, 9}, is 8. Therefore, for a finite
number of observations, we can easily find the mode. A set of values may
have one mode or more than one mode or no mode at all.
43. Definition:
• A mode is defined as the value that has a higher frequency in a given set of
values. It is the value that appears the most number of times.
• Example: In the given set of data: 2, 4, 5, 5, 6, 7, the mode of the data set is
5 since it has appeared in the set twice.
44. Bimodal, Trimodal & Multimodal (More than one
mode)
• When there are two modes in a data set, then the set is called bimodal
• For example, The mode of Set A = {2,2,2,3,4,4,5,5,5} is 2 and 5, because
both 2 and 5 is repeated three times in the given set.
• When there are three modes in a data set, then the set is called trimodal
• For example, the mode of set A = {2,2,2,3,4,4,5,5,5,7,8,8,8} is 2, 5 and 8
• When there are four or more modes in a data set, then the set is
called multimodal
45. Solution:
• The value occurring most frequently in a set of observations is its mode. In other words, the
mode of data is the observation having the highest frequency in a set of data. There is a
possibility that more than one observation has the same frequency, i.e. a data set could have
more than one mode. In such a case, the set of data is said to be multimodal.
• Let us look into an example to get a better insight.
• Example: The following table represents the number of wickets taken by a bowler in 10
matches. Find the mode of the given set of data.
•
• It can be seen that 2 wickets were taken by the bowler frequently in different matches.
Hence, the mode of the given data is 2.
46. Mode Formula For Grouped Data:
• In the case of grouped frequency distribution, calculation of mode just by
looking into the frequency is not possible. To determine the mode of data in
such cases we calculate the modal class. Mode lies inside the modal class. The
mode of data is given by the formula:
47. • Where,
• l = lower limit of the modal class
• h = size of the class interval
• f1 = frequency of the modal class
• f0 = frequency of the class preceding the modal class
• f2 = frequency of the class succeeding the modal class
48. Solution:
• Let us learn here how to find the mode of a given data with the help of examples.
Example 1: Find the mode of the given data set: 3, 3, 6, 9, 15, 15, 15, 27, 27, 37, 48.
Solution: In the following list of numbers,
3, 3, 6, 9, 15, 15, 15, 27, 27, 37, 48
15 is the mode since it is appearing more number of times in the set compared to other numbers.
Example 2: Find the mode of 4, 4, 4, 9, 15, 15, 15, 27, 37, 48 data set.
Solution: Given: 4, 4, 4, 9, 15, 15, 15, 27, 37, 48 is the data set.
As we know, a data set or set of values can have more than one mode if more than one value
occurs with equal frequency and number of time compared to the other values in the set.
Hence, here both the number 4 and 15 are modes of the set.
49. Example :
• In a class of 30 students marks obtained by students in mathematics out of
50 is tabulated as below. Calculate the mode of data given.
50. Solution:
• The maximum class frequency is 12 and the class interval corresponding to this
frequency is 20 – 30. Thus, the modal class is 20 – 30.
• Lower limit of the modal class (l) = 20
• Size of the class interval (h) = 10
• Frequency of the modal class (f1) = 12
• Frequency of the class preceding the modal class (f0) = 5
• Frequency of the class succeeding the modal class (f2)= 8
• Substituting these values in the formula we get;
51.
52. Standard Deviation
• The spread of statistical data is measured by the standard deviation.
Distribution measures the deviation of data from its mean or average
position. The degree of dispersion is computed by the method of estimating
the deviation of data points. It is denoted by the symbol, ‘σ’.
• The standard deviation is then defined as the positive square root of the
arithmetic mean of the squares of the deviations taken from the arithmetic
mean.
53. Merits of Standard Deviation
• It is rigidly defined
• It is based on all observations
• It does not ignore the algebraic signs of deviations
• It is capable of further mathematical treatment
• It is not much affected by sampling fluctuations.
54. Demerits of Standard Deviation
• It is difficult to understand and calculate
• It cannot be calculated for qualitative data and distribution with open end
classes.
• It is unduly affected due to extreme deviations.