STATS
A frequency distribution is a summary of the frequency of occurrences of values within a dataset. It
organizes data into groups or intervals and counts the number of occurrences in each group. This is
particularly useful when dealing with large datasets or continuous variables where individual data points
may not be as meaningful on their own.
Here's an example to illustrate:
Let's say you have a dataset of exam scores for a class of students:
78,82,65,92,78,82,88,72,65,82,78,88,92,78,8278,82,65,92,78,82,88,72,65,82,78,88,92,78,82
To create a frequency distribution from this dataset, you first need to determine the unique values (or
intervals if dealing with continuous data) and then count how many times each value occurs.
1. Unique Values:
65, 72, 78, 82, 88, 92
2. Frequency Count:
65: 2 times
72: 1 time
78: 4 times
82: 4 times
88: 2 times
92: 2 times
Now, you can represent this information in a table format:
Score Frequency
65 2
72 1
78 4
82 4
88 2
92 2
From this frequency distribution, you can easily see the distribution of scores within the dataset. For
example, you can see that scores of 78 and 82 occurred most frequently (4 times each), while scores of
72 occurred only once. This kind of summary is helpful for understanding the distribution of data and
identifying patterns or outliers.
Mean, median, and mode are three measures of central tendency used in statistics to describe the
center of a data set. Here's an explanation of each with examples:
1. Mean: The mean, often referred to as the average, is calculated by summing up all the values in
a data set and then dividing the sum by the total number of values.
Example: Let's say we have a data set representing the scores of 5 students on a test: {85,92,78,90,88}
{85,92,78,90,88} To find the mean, we sum up all the scores and divide by the total number of scores:
Mean=85+92+78+90+885=4335=86.6Mean=585+92+78+90+88=5433=86.6 So, the mean score is 86.6.
2. Median: The median is the middle value of a data set when the values are arranged in ascending
or descending order. If there is an even number of values, the median is the average of the two
middle values.
Example: Let's consider the same data set of test scores: {85,92,78,90,88}{85,92,78,90,88} First, we
need to arrange the scores in ascending order: {78,85,88,90,92}{78,85,88,90,92} Since there's an odd
number of values (5), the median is the middle value, which is 88. So, the median score is 88.
3. Mode: The mode is the value that appears most frequently in a data set. A data set can have
one mode, more than one mode (multimodal), or no mode if all values occur with the same
frequency.
Example: Consider the following data set representing the number of goals scored by a football team in
10 matches: {2,1,3,2,4,2,1,2,3,1}{2,1,3,2,4,2,1,2,3,1} In this data set, the value 2 appears most frequently
(four times), so the mode is 2. Therefore, the mode is 2.
In summary, mean represents the average value, median represents the middle value, and mode
represents the most frequently occurring value in a data set. Each measure has its own utility depending
on the nature of the data and the context of analysis.
Standard deviation is a measure of the amount of variation or dispersion in a set of values. It tells you
how spread out the values in a data set are from the mean. A low standard deviation indicates that the
data points tend to be close to the mean, while a high standard deviation indicates that the data points
are spread out over a wider range of values.
Here's how to calculate the standard deviation, along with an example:
1. Calculate the Mean: First, find the mean (average) of the data set.
2. Calculate the Variance: Subtract the mean from each data point, square the result, and then
find the average of those squared differences.
3. Take the Square Root: Finally, take the square root of the variance to find the standard
deviation.
Let's illustrate this with an example:
Example: Suppose we have a data set representing the ages of 5 people: {25,30,35,40,45}
{25,30,35,40,45}
1. Calculate the Mean: Mean=25+30+35+40+455=1755=35Mean=525+30+35+40+45=5175=35
2. Calculate the Variance: Subtract the mean from each data point, square the result, and find the
average of those squared differences. Squared Differences: {(25−35)2,(30−35)2,(35−35)2,
(40−35)2,(45−35)2}Squared Differences: {(25−35)2,(30−35)2,(35−35)2,(40−35)2,(45−35)2}
={(−10)2,(−5)2,(0)2,(5)2,(10)2}={(−10)2,(−5)2,(0)2,(5)2,(10)2}
={100,25,0,25,100}={100,25,0,25,100}
Now, find the average of these squared differences:
Variance=100+25+0+25+1005=2505=50Variance=5100+25+0+25+100=5250=50
3. Take the Square Root: Finally, take the square root of the variance to find the standard
deviation. Standard Deviation=50≈7.07Standard Deviation=50≈7.07
So, the standard deviation of the ages in this data set is approximately 7.07 years. This means that the
ages are, on average, about 7.07 years away from the mean age of 35.
