11.2 DEFINITION AND CALCULATION OF MEAN, MEDIAN AND MODE

In our daily life routine, we commonly hear the somewhat similar to following statements. i. Heartbeat of human is 72 beats per minute ii. Death rate due to cancer in Pakistan is 25 per thousand patients. iii. Production of wheat in Punjab is 2000 Kg per acer. iv. The price of the bananas in the market is Rs. 120 per dozen. v. Food consumption of a human is 1kg per day. vi. The rain fall in Islamabad is 1500mm per year. If we think on about above mentioned statements, none of them is found exactly correct. In statement no. "i", our heart may not be same 72 in each minute of the day. During running it may be 90 per minute and during sleep it may be 60 per minute. Our heartbeat is approximately 72 in each minute. As given in statement "ii" It is quite possible that in one hospital only 5 cancer patient died out of 1000 visited for treatment and in other hospital 60 patients died out of 1000 patients gone through treatment. The death rate of cancer patients is about 25 per 1000, may not be same all the year and in every hospital. The production of wheat in one district of Punjab may be 1500 Kg from an acer and it may be 2400 Kg from one acer in another district. Although above statements are not exactly true but still they are very important. Actually, these are approximate statements in specific situations. In terms of statistics, we call such statements as average statements. In our daily conversation, we make many statements which have some meaning only on average basis. In different fields of life like agriculture, health, poultry, the idea of average is very important. Many experts at national and international level discuss the findings of studies in averages. The average is also called measure of central tendency.

Calculation of Average: Average is a single value which is calculated to represent the whole data. It may be calculated for a data sample of patients or a population of migrating birds etc. The average is a value which expresses the central idea of the observations. There are different ways to represent average of a data in different situations. For representation of a specific data, proper type of average is used by the expert who is calculating the average.

Types of averages: The following types of averages are commonly used: (i) Arithmetic Mean (ii) Median (iii) Mode (iv) Geometric mean (v) Harmonic mean Here in this chapter we will study only first three types of averages in detail.

11.2.1 Arithmetic Mean or simply Mean

Definition: Mean is the sum of all the values of data set divided by the total number of values in the data set. It is the single value which is calculated to represent the whole set of data. The symbol "x" (read as "x bar") represent the sample mean. The bar above the letter x represents the mean of a set of values:

Example 1(un-group data)

Dataset: Team of world health organization (WHO) planned to assess the prevalence of polio disease in Pakistan. Study to record the polio cases in Pakistan continued for consecutive two years. Number of confirmed Polio patients detected each month of 2022 and 2023 are given in table below. Calculate the mean polio patients per month in each year. Also compare the prevalence of Polio disease in both years.

Mean number of Polio patients per month in 2022 $=$ $\frac{\mathrm{S}\mathrm{u}\mathrm{m}\mathrm{o}\mathrm{f}\mathrm{P}\mathrm{o}\mathrm{l}\mathrm{i}\mathrm{o}\mathrm{c}\mathrm{a}\mathrm{s}\mathrm{e}\mathrm{s}\mathrm{i}\mathrm{n}\mathrm{c}\mathrm{o}\mathrm{m}\mathrm{p}\mathrm{l}\mathrm{e}\mathrm{t}\mathrm{e}\mathrm{y}\mathrm{e}\mathrm{a}\mathrm{r}}{\mathrm{T}\mathrm{o}\mathrm{t}\mathrm{a}\mathrm{l}\mathrm{n}\mathrm{u}\mathrm{m}\mathrm{b}\mathrm{e}\mathrm{r}\mathrm{o}\mathrm{f}\mathrm{M}\mathrm{o}\mathrm{n}\mathrm{t}\mathrm{h}\mathrm{s}}=\frac{192}{12}=16$$\frac{\mathrm{Sum~of~Polio~cases~in~complete~year}}{\mathrm{Total~number~of~Months}} = \frac{192}{12} = 16$

Mean number of Polio patients per month in 2023 $=$ $\frac{\mathrm{S}\mathrm{u}\mathrm{m}\mathrm{o}\mathrm{f}\mathrm{P}\mathrm{o}\mathrm{l}\mathrm{i}\mathrm{o}\mathrm{c}\mathrm{a}\mathrm{s}\mathrm{e}\mathrm{s}\mathrm{i}\mathrm{n}\mathrm{c}\mathrm{o}\mathrm{m}\mathrm{p}\mathrm{l}\mathrm{e}\mathrm{t}\mathrm{e}\mathrm{y}\mathrm{e}\mathrm{a}\mathrm{r}}{\mathrm{T}\mathrm{o}\mathrm{t}\mathrm{a}\mathrm{l}\mathrm{n}\mathrm{u}\mathrm{m}\mathrm{b}\mathrm{e}\mathrm{r}\mathrm{o}\mathrm{f}\mathrm{M}\mathrm{o}\mathrm{n}\mathrm{t}\mathrm{h}\mathrm{s}}=\frac{144}{12}=12$$\frac{\mathrm{Sum~of~Polio~cases~in~complete~year}}{\mathrm{Total~number~of~Months}} = \frac{144}{12} = 12$

Difference of Polio patients per month detected in 2022 and $2023 = 16 - 12 = 04$ Prevalence of Polio disease decreased in 2023 compared to 2022 by 04 patients per month.

11.2.2. Median

Definition: The median is the middle value in a dataset that are arranged in ascending or descending order of magnitude. Median divides the data set into two equal parts. One part of the dataset have the values less than the middle value and the other part of the dataset have values greater than the middle item. It is denoted by $x$ , (read as "x tilde" or "x snake"). It is necessary to arrange the values of dataset in ascending or descending order.

The methods of calculating the median are simple and the value of median is not affected by change in extreme values of dataset.

Median for un-group data

If the number of values is odd, the median is the middle value.

Median $(x = 0)$ The value of ${\left(\frac{n+1}{2}\right)}^n$$\left(\frac{n + 1}{2}\right)^{n}$ item in the dataset (where $n =$ number of items or data) If the number of values is even, the median is the average of the two middle values.

The measured heights of plants after arranging in ascending order is as given below: 46,49,55,58,62,63,64,65,66,67,68,70,72 Number of values in the dataset of observed heights $= 13$ Median $(x)=\frac{n+1}{2}=\frac{13+1}{2}=\frac{14}{2}=7$$(x) = \frac{n + 1}{2} = \frac{13 + 1}{2} = \frac{14}{2} = 7$ (i.e. value of $7^{\mathrm{t}\mathrm{h}}$$7^{\mathrm{th}}$ value in dataset) Median $(x) = 64$

Example 2: (For the dataset with even number of values)

Dataset: The number of fruits produced on each of the 16 plants of experimental group are recorded as below:

3,5,18,21,15,10,8,12,13,7,11,14,9,16,20,24 Find out the median value of fruits produced per plant.