Median

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Median

Median is the middle number in a sorted list of numbers. To determine the median value in a sequence of numbers, the numbers must first be arranged in value order from lowest to highest. If there is an odd amount of numbers, the median value is the number that is in the middle, with the same amount of numbers below and above. If there is an even amount of numbers in the list, the middle pair must be determined, added together and divided by two to find the median value. The median can be used to determine an approximate Average, or mean. The median is sometimes used as opposed to the mean when there are outliers in the sequence that might skew the average of the values. The median of a sequence can be less affected by outliers than the mean.  

To find the median value in a list with an odd amount of numbers, one would find the number that is in the middle with an equal amount of numbers on either side of the median.

Example type 1

To find the median, first arrange the numbers in order from lowest to highest: List: 3, 13, 2, 34, 11, 26, 47  

Arranged in order, the list becomes: 2, 3, 11, 13, 26, 34, 47

The median is the number in the middle: 2, 3, 11, 13, 26, 34

13 is the median in the list of numbers since there are 3 numbers on either side.  

Example type 2

with an even amount of numbers things are slightly different.

In that case we find the middle pair of numbers, and then find the value that is half way between them. This is easily done by adding them together and dividing by two.

Suppose 3, 13, 7, 5, 21, 23, 23, 40, 23, 14, 12, 56, 23, 29 is the number series

When we put those numbers in order we have:  3, 5, 7, 12, 13, 14, 21, 23, 23, 23, 23, 29, 40, 56

There are now fourteen numbers and so we don’t have just one middle number, we have a pair of middle numbers:

3, 5, 7, 12, 13, 14, 21, 23, 23, 23, 23, 29, 40, 56

middle numbers are 21 and 23.

To find the value halfway between them, add them together and divide by 2:  

21 + 23 = 44 then 44 ÷ 2 = 22

So the Median in this example is 22.

Mode

Mode is the value which occurs most frequently in a set of observations. Simply put, it is the number which is repeated most, i.e. the number with the highest frequency. In the field of statistics, it is an important tool to interpret data in a relevant manner. Now it is possible for the data set to be multimodal (have more than one mode) which means more than one observation has the same number of frequencies.

Example: Let us find the Mode of the following data

4, 89, 65, 11, 54, 11, 90, 56

Here in these varied observations the most occurring number is 11, hence the Mode = 11

Mode of Grouped Data

As we know that Mode is the most frequently occurring number of a data set. This is easily recognizable in an ungrouped dataset, but if the data set is presented in class intervals, this can get a bit tricky. So how can we calculate Mode of grouped data?

Solution method

• Create a table with two columns
• In column 1 write your class intervals
• In column 2 write the corresponding frequencies
•  Locate the maximum frequency denoted by fm
• Determine the class corresponding to fm , this will be your Modal class
• Calculate the Mode using given formula  

Mode = L +fm−f1 /  (2fm−f1−f2) × h

Where,

L = lower limit of Modal Class

f_m = frequency of modal class

h = width of modal class

f₁ = frequency of pre modal class

f₂ = frequency of post modal class

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The median is a measure of central tendency, describing the middle value in a sorted, ascending or descending, list of numbers and can be more descriptive than the arithmetic mean when the data set is skewed. The median of a list of numbers can be found by arranging the numbers in order, and then finding the ‘middle’ number. If there is an even number of numbers, the median is the mean of the two middle numbers.

The median is a robust measure of central tendency, meaning that it is not as sensitive to outliers as the mean. This makes it a good choice for data sets that may have a few extreme values.

The median can be used to compare two or more data sets. For example, if you want to compare the salaries of men and Women, you could calculate the median salary for each group. If the median salary for men is higher than the median salary for women, this would suggest that men are, on average, paid more than women.

The median can also be used to identify outliers. An outlier is a data point that is significantly different from the other data points in the set. Outliers can be identified by calculating the median and then looking for data points that are more than 1.5 times the interquartile range away from the median.

The median is a useful measure of central tendency, but it is important to be aware of its limitations. The median is not as sensitive to changes in the data as the mean, so it may not be the best choice for data sets that are not normally distributed. Additionally, the median can be difficult to calculate for data sets that contain a large number of ties.

Despite its limitations, the median is a valuable tool for data analysis. It is a robust measure of central tendency that is not as sensitive to outliers as the mean. The median can be used to compare two or more data sets, identify outliers, and make inferences about the Population from which the data was collected.

Here are some examples of how the median can be used:

• To find the median age of a group of people, you would arrange their ages in order from youngest to oldest and then find the middle age.
• To find the median income of a group of households, you would arrange their incomes in order from lowest to highest and then find the middle income.
• To find the median score on a test, you would arrange the scores in order from lowest to highest and then find the middle score.

The median can also be used to calculate the interquartile range (IQR). The IQR is a measure of variability that is not as sensitive to outliers as the standard deviation. To calculate the IQR, you first find the median of the data set. Then, you find the median of the lower half of the data set and the median of the upper half of the data set. The IQR is the difference between these two medians.

The median can be used to make inferences about the population from which the data was collected. For example, if you know that the median income of a group of people is $50,000, you can infer that half of the people in the group earn less than $50,000 and half of the people in the group earn more than $50,000.

The median is a valuable tool for data analysis. It is a robust measure of central tendency that is not as sensitive to outliers as the mean. The median can be used to compare two or more data sets, identify outliers, and make inferences about the population from which the data was collected.

Mode

• What is the mode?
  The mode is the most frequent number in a set of data.

• How do you find the mode?
  To find the mode, you simply count how many times each number appears in the set of data. The number that appears the most times is the mode.

• What are some examples of modes?
  Some examples of modes are:

  • The mode of the set {1, 2, 3, 3, 3, 4} is 3.
  • The mode of the set {5, 5, 5, 6, 7} is 5.
  • The mode of the set {1, 2, 3, 4, 5} is not a single number, because each number appears once.

• What are some properties of the mode?
  The mode is a robust measure of central tendency, meaning that it is not affected by outliers. The mode is also easy to calculate and understand. However, the mode can be misleading if there are multiple modes in a set of data.

Mean

• What is the mean?
  The mean is the average of a set of numbers. To find the mean, you add all of the numbers in the set and then divide by the number of numbers in the set.

• How do you find the mean?
  To find the mean, you simply add all of the numbers in the set and then divide by the number of numbers in the set. For example, to find the mean of the set {1, 2, 3, 4}, you would add 1 + 2 + 3 + 4 = 10 and then divide by 4 = 2.5.

• What are some examples of means?
  Some examples of means are:

  • The mean of the set {1, 2, 3, 3, 4} is 2.8.
  • The mean of the set {5, 5, 5, 6, 7} is 5.8.
  • The mean of the set {1, 2, 3, 4, 5} is 3.

• What are some properties of the mean?
  The mean is a measure of central tendency that is often used in statistics. The mean is also known as the arithmetic mean. The mean is easy to calculate and understand, but it can be sensitive to outliers.

Median

• What is the median?
  The median is the middle number in a set of numbers arranged in order from least to greatest. If there are two middle numbers, the median is the average of those two numbers.

• How do you find the median?
  To find the median, you first arrange the numbers in the set in order from least to greatest. If there are an even number of numbers in the set, the median is the average of the two middle numbers. If there are an odd number of numbers in the set, the median is the middle number.

• What are some examples of medians?
  Some examples of medians are:

  • The median of the set {1, 2, 3, 3, 4} is 3.
  • The median of the set {5, 5, 5, 6, 7} is 5.
  • The median of the set {1, 2, 3, 4, 5} is 3.

• What are some properties of the median?
  The median is a measure of central tendency that is often used in statistics. The median is also known as the second quartile. The median is not sensitive to outliers, but it can be difficult to calculate if there are a large number of numbers in the set.

Sure, here are some multiple choice questions about the topics of mean, mode, and range without mentioning the topic of median:

1. Which of the following is a measure of central tendency?
   (A) Mean
   (B) Mode
   (C) Range
   (D) All of the above

2. The mean is calculated by adding up all the values in a set and then dividing by the number of values.
   (A) True
   (B) False

3. The mode is the most frequent value in a set.
   (A) True
   (B) False

4. The range is the difference between the largest and smallest values in a set.
   (A) True
   (B) False

5. A set of data has a mean of 5, a mode of 3, and a range of 8. Which of the following is true?
   (A) The median is 5.
   (B) The median is 3.
   (C) The median is 8.
   (D) Cannot be determined from the information given.

6. A set of data has a mean of 5, a mode of 3, and a median of 4. Which of the following is true?
   (A) The range is 8.
   (B) The range is 2.
   (C) The range is 1.
   (D) Cannot be determined from the information given.

7. A set of data has a mean of 5, a mode of 3, and a median of 4. Which of the following is true?
   (A) The data is skewed to the left.
   (B) The data is skewed to the right.
   (C) The data is symmetrical.
   (D) Cannot be determined from the information given.

8. A set of data has a mean of 5, a mode of 3, and a median of 4. Which of the following is true?
   (A) The data is normally distributed.
   (B) The data is not normally distributed.
   (C) Cannot be determined from the information given.

9. A set of data has a mean of 5, a mode of 3, and a median of 4. Which of the following is true?
   (A) The data is consistent.
   (B) The data is not consistent.
   (C) Cannot be determined from the information given.

10. A set of data has a mean of 5, a mode of 3, and a median of 4. Which of the following is true?
    (A) The data is complete.
    (B) The data is not complete.
    (C) Cannot be determined from the information given.