## Mean, Median, Mode Pharmaceutical examples

**Biostatistics and research methodology notes Unit 1**: Introduction Statistics, Biostatistics, Frequency distribution Measures of central tendency Mean, Median, Mode- Pharmaceutical examples Measures of dispersion Dispersion, Range, standard deviation, Pharmaceutical problems Correlation Definition, Karl Pearson’s coefficient of correlation, Multiple correlations – Pharmaceuticals examples

**Mean**

The “average” number; is found by adding all data points and dividing by the number of data points.

Mean = (Sum of all the observations/Total number of observations) |

The basic formula to calculate the mean is calculated based on the given data set. Each term in the data set is considered while evaluating the mean. The general formula for mean is given by the ratio of the sum of all the terms and the total number of terms. Hence, we can say;

**Mean = Sum of the Given Data/Total number of Data**

To calculate the arithmetic mean of a set of data we must first add up (sum) all of the data values (x) and then divide the result by the number of values (n). Since ∑ is the symbol used to indicate that values are to be summed (see Sigma Notation) we obtain the following formula for the mean (x̄):

**x̄=∑ x/n**

**Example:**

What is the mean of 2, 4, 6, 8 and 10?

**Solution:**

First, add all the numbers.

2 + 4 + 6 + 8 + 10 = 30

Now divide by 5 (total number of observations).

Mean = 30/5 = 6

In the case of a discrete probability distribution of a random variable X, the mean is equal to the sum over every possible value weighted by the probability of that value; that is, it is computed by taking the product of each possible value x of X and its probability P(x) and then adding all these products together.

**Median**

The middle number; found by ordering all data points and picking out the one in the middle (or if there are two middle numbers, taking the mean of those two numbers).

Median, in statistics, is the middle value of the given list of data, when arranged in an order. The arrangement of data or observations can be done either in ascending order or descending order.

Example: The median of 2,3,4 is 3

**Example 1:**

Find the Median of 14, 63 and 55

**solution:**

Put them in ascending order: 14, 55, 63

The middle number is 55, so the median is 55.

**Example 2:**

Find the median of the following:

4, 17, 77, 25, 22, 23, 92, 82, 40, 24, 14, 12, 67, 23, 29

**Solution:**

When we put those numbers in the order we have:

4, 12, 14, 17, 22, 23, 23, 24, 25, 29, 40, 67, 77, 82, 92,

There are fifteen numbers. Our middle is the eighth number:

The median value of this set of numbers is 24.

**Example 3:**

Rahul’s family drove through 7 states on summer vacation. The prices of Gasoline differ from state to state. Calculate the median of gasoline cost.

1.79, 1.61, 2.09, 1.84, 1.96, 2.11, 1.75

**Solution:**

By organizing the data from smallest to greatest, we get:

1.61, 1.75, 1.79, **1.84** , 1.96, 2.09, 2.11

Hence, the median of the gasoline cost is 1.84. There are three states with greater gasoline costs and 3 with smaller prices.

**What is the difference between mean and median?**Median is defined as the center value of ordered list of values.

Mean is the ratio of sum of list of values and number of values, order of values does not matter.

**Mode**

The most frequent number that is, the number that occurs the highest number of times.

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.

Statistics deals with the presentation, collection and analysis of data and information for a particular purpose. We use tables, graphs, pie charts, bar graphs, pictorial representation, etc. After the proper organization of the data, it must be further analyzed to infer helpful information.

For this purpose, frequently in statistics, we tend to represent a set of data by a representative value that roughly defines the entire data collection. This representative value is known as the measure of central tendency. By the name itself, it suggests that it is a value around which the data is centred. These measures of central tendency allow us to create a statistical summary of the vast, organized data. One such measure of central tendency is the mode of data.

The most frequent number occurring in the data set is known as the mode.

Consider the following data set which represents the marks obtained by different students in a subject.

Name | Anmol | Kushagra | Garima | Ashwini | Geetika | Shakshi |

Marks Obtained (out of 100) | 73 | 80 | 73 | 70 | 73 | 65 |

The maximum frequency observation is 73 ( as three students scored 73 marks), so the mode of the given data collection is 73.

**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.

**Example 3: Find the mode of 3, 6, 9, 16, 27, 37, 48.**

**Solution:** If no value or number in a data set appears more than once, then the set has no mode.

Hence, for set 3, 6, 9, 16, 27, 37, 48, there is no mode available.

## Pharmaceutical examples

**Problem 1**– **Weights in mg of tablets in a sample data are 110, 122, 109, 121, 105 Find out the Sample mean?**

X is the summation of the tablet

weight i.e. 110+125+100+120+105

N is the summation of the number of

observations

i.e = 5

110+125+100+120+ 105=160

160/5=32 (Answer- The sample mean for the above problem is 32)

**Problem 2: Calculate the mean for the following data on systolic BP of volunteers.**

**Solution**

- This data is an example of discrete series data.
- In this data Frequency (f) and Sample size (x) were given.
- In each case multiply f and x
- Summarize fx value.
- Summarize f values

**Problem 3: Calculate the mean for the following data on systolic BP of volunteers.**

**Solution**

- This data is an example of continuous series data, where the class interval is given.
- Determine the Midpoint of the data and considered it as ( x ).
- Midpoint is determined in each case by adding lower limit and upper limit divided by 2.
- In each case multiply f and x
- Summarize fx value.
- Summarize f values

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