STANDARD DEVIATION AND VARIATION

Biostatistics Biostatistics is a branch of statistics that applies statistical methods to biological, medical, and public health research. It's a crucial field in analyzing data from experiments, clinical trials, epidemiological studies, genetics, and more. Key Areas of Biostatistics: 1. Descriptive Statistics o Summarizing and organizing data (e.g., means, medians, standard deviations, histograms). 2. Inferential Statistics o Making predictions or generalizations about a population based on a sample (e.g., hypothesis testing, confidence intervals, p-values). 3. Probability Theory o Understanding and modeling random phenomena, crucial for designing experiments and interpreting data. 4. Study Design o Includes randomized controlled trials, cohort studies, case-control studies, and cross-sectional studies. 5. Regression and Modeling o Linear and logistic regression, survival analysis (Cox proportional hazards), mixed-effects models, etc. 6. Survival Analysis o Analyzing time-to-event data, often used in clinical trials (e.g., time until disease progression or death). 7. Longitudinal Data Analysis o Involves data collected over time from the same subjects, requiring specialized models like mixed-effects models. 8. Statistical Genetics and Bioinformatics o Analyzing data from genetic studies, including genome-wide association studies (GWAS), sequencing data, etc. 9. Bayesian Methods o Incorporating prior information into statistical analyses, especially useful in small sample studies. Applications: • Designing and analyzing clinical trials • Evaluating risk factors for diseases • Monitoring public health trends • Genetic mapping and genomics • Health economics and outcomes research MEAN,MEDIAN, MODE The difference between mean, median and mode are: • Mean is the average value of the given observations • Median is the middle value of the given observations • Mode is the most repeated value in the given observation What is Mean, Median, and Mode? The mean is the average where the sum of all the numbers is divided by the total number of numbers, whereas the median is the middle value in the list of given numbers numerically ordered from smallest to biggest and mode is the value of the number which occurs most often in the list. You can learn more about it here: What are the differences between Mean, Median, and Mode? These three terms are related to each other. There’s a relationship between mean, median and mode and is called an empirical relationship between them. Below are some of the most integral differences between the mean, median and mode. Sl. No. Mean Median Mode 1. The average taken for a set of numbers is called a mean. The middle value in the data set is called the Median. The number that occurs the most in a given list of numbers is called a mode. 2. Add all of the numbers together and divide the sum by the total number of values. Place all the given numbers in an ascending order It shows the frequency of occurrence. 3. The result is the mean or average score. The next step is to find the middle number on the list. It is called the median. We can have more than one mode or no mode at all. 4. Example: To find the average of the four numbers 2, 4, 6, and 8, we need to add the number first. • 2 + 4 + 6+ 8 = 20 • Divide the sum by the total number of numbers, i. e 4. • 20/4 = 5 is the average or mean Example: 4, 2, 8, 10, 19. • Arrange the numbers in ascending order. i .e., 2, 4, 8, 10, 19. • As the total numbers are 5, so the middle number 8 is the median here. Example: 3, 3, 5, 6, 7, 7, 8, 1, 1, 1, 4, 5, 6. • Find the frequency of each number. • For number 3, it’s 2. For 5, it’s 2. For 6, it’s 2. For 7, it’s 2. For 8, it’s one. For 1, it’s 3. For 4, it’s 1. • The number with the highest frequency is the mode. Hence, the mode of the given sequence of numbers is 1. Q1)What are the three measures of central tendencies? The three measures of central tendencies are mean, median and mode. Q2)What is mean in statistics? Give an example. The mean is the average of given data values. The formula to calculate the mean value is: Mean = Sum of observation/Number of observations For example, Mean of 2, 5, 6, 7, 8 is, (2+5+6+7+8)/5 = 28/5 = 5.6 Q3)What is a median? Explain with an example. The median is the middle value of a given observation. For example, 23, 33, 43, 63, and 53 is a set of observations; then, to find the median, we need to arrange the given values in an order (ascending or descending). Hence, we get 23, 33, 43, 53, 63 Therefore, Median = 43 Q4)What is a mode? Give an example. Mode represents the value which is repeated the maximum number of times in a given set of observations. For example, 11, 12, 13, 13, 14, and 15 are the set of data. Here, the number 13 is repeated twice and is considered to be the mode value. Q5)What are the mean, median and mode formulas? Mean = Sum of observation/Number of observation Median = {(n+1)/2}th term when n is odd & Median = [(n/2)th term + {(n/2)+1}th]/2 when n is even Mode = Value repeated the maximum number of times. What are the Variance and Standard Deviation? In statistics, Variance and standard deviation are related with each other since the square root of variance is considered the standard deviation for the given data set. Below are the definitions of variance and standard deviation. What is variance? Variance is the measure of how notably a collection of data is spread out. If all the data values are identical, then it indicates the variance is zero. All non-zero variances are considered to be positive. A little variance represents that the data points are close to the mean, and to each other, whereas if the data points are highly spread out from the mean and from one another indicates the high variance. In short, the variance is defined as the average of the squared distance from each point to the mean. What is Standard deviation? Standard Deviation is a measure which shows how much variation (such as spread, dispersion, spread,) from the mean exists. The standard deviation indicates a “typical” deviation from the mean. It is a popular measure of variability because it returns to the original units of measure of the data set. Like the variance, if the data points are close to the mean, there is a small variation whereas the data points are highly spread out from the mean, then it has a high variance. Standard deviation calculates the extent to which the values differ from the average. Standard Deviation, the most widely used measure of dispersion, is based on all values. Therefore a change in even one value affects the value of standard deviation. It is independent of origin but not of scale. It is also useful in certain advanced statistical problems. Variance and Standard Deviation Formula The formulas for the variance and the standard deviation is given below: Standard Deviation Formula The population standard deviation formula is given as: σ=1N∑i=1N(Xi−μ)2 Here, σ = Population standard deviation N = Number of observations in population Xi = ith observation in the population μ = Population mean Similarly, the sample standard deviation formula is: s=1n−1∑i=1n(xi−x―)2 Here, s = Sample standard deviation n = Number of observations in sample xi = ith observation in the sample x― = Sample mean Variance Formula: The population variance formula is given by: σ2=1N∑i=1N(Xi−μ)2 The sample variance formula is given by: s2=1n−1∑i=1n(xi−x―)2 How is Standard Deviation calculated? The formula for standard deviation makes use of three variables. The first variable is the value of each point within a data set, with a sum-number indicating each additional variable (x, x1, x2, x3, etc). The mean is applied to the values of the variable M and the number of data that is assigned to the variable n. Variance is the average of the values of squared differences from the arithmetic mean. To calculate the mean value, the values of the data elements have to be added together and the total is divided by the number of data entities that were involved. Standard deviation, denoted by the symbol σ, describes the square root of the mean of the squares of all the values of a series derived from the arithmetic mean which is also called the root-mean-square deviation. 0 is the smallest value of standard deviation since it cannot be negative. When the elements in a series are more isolated from the mean, then the standard deviation is also large. The statistical tool of standard deviation is the measures of dispersion that computes the erraticism of the dispersion among the data. For instance, mean, median and mode are the measures of central tendency. Therefore, these are considered to be the central first order averages. The measures of dispersion that are mentioned directly over are averages of deviations that result from the average values, therefore these are called second-order averages.

Structure of bacteria One of the very first organisms to evolve on earth was probably a unicellular organism, similar to modern bacteria. Ever since then, life has evolved into a multitude of life forms over many millennia. However, we can still trace our ancestry back to this single-celled organism. Bacteria Definition “Bacteria are unicellular organisms belonging to the prokaryotic group where the organisms lack a few organelles and a true nucleus”. Bacteria Diagram The bacteria diagram given below represents the structure of a typical bacterial cell with its different parts. The cell wall, plasmid, cytoplasm and flagella are clearly marked in the diagram.

Ultrastructure of a Bacteria Cell The structure of bacteria is known for its simple body design. Bacteria are single-celled microorganisms with the absence of the nucleus and other cell organelles; hence, they are classified as prokaryotic organisms. They are also very versatile organisms, surviving in extremely inhospitable conditions. Such organisms are called extremophiles. Extremophiles are further categorized into various types based on the types of environments they inhabit: 1. Thermophiles 2. Acidophiles 3. Alkaliphiles 4. Osmophiles 5. Barophiles 6. Cryophiles Another fascinating feature of bacteria is their protective cell wall, which is made up of a special protein called peptidoglycan. The components of bacterial cell wall forms an important basis upon which the bacteria can be divided. This particular protein isn’t found anywhere else in nature except in the cell walls of bacteria. But few of them are devoid of this cell wall, and others have a third protection layer called capsule. On the outer layer, one or more flagella or pili is attached, and it functions as a locomotory organ. Pili can also help certain bacteria to attach themselves to the host’s cells. They do not contain any cell organelle as in animal or plant cell except for ribosomes. Ribosomes are the sites of protein synthesis. In addition to this DNA, they have an extra circular DNA called plasmid. These plasmids make some strains of bacteria resistant to antibiotics. BACTERIAL GROWTH Bacterial Growth Bacteria are unicellular organisms that tend to reproduce asexually by the means of binary fission. Bacterial growth is the increase in the number of bacterial cells rather than the increase in their cell size. The growth of these bacterial cells takes place in an exponential manner, i.e., one cell divides into 2, then 4, then 8, 16, 32 and so on. The time taken for a bacterial cell to double is called generation time. The generation time varies among different species of bacteria based on the environmental conditions they grow in. Clostridium perfringens is the fastest growing bacteria that has a generation time of 10 minutes while Escherichia coli has a doubling time of 20 minutes. Mycobacterium tuberculosis is one of the slowest growing bacteria, taking about 12 to 16 hours to double. Growth Curve In a closed system with enough nutrients, a bacteria shows a predictable growth pattern that is the bacterial growth curve. It consists of four different phases. Read on to learn about the phases in detail. Phases of the Bacterial Growth Curve Upon inoculation into a new nutrient medium, the bacteria shows four distinct phases of growth. Let us dive into each of the phases in detail – Lag Phase The bacteria upon introduction into the nutrient medium take some time to adapt to the new environment. In this phase, the bacteria does not reproduce but prepares itself for reproduction. The cells are active metabolically and keep increasing in size. The cells synthesise RNA, growth factors and other molecules required for cell division. Log Phase Soon after the lag phase, i.e., the preparation phase, the bacterial cells enter the log phase. The log phase is also known as the exponential phase. This phase is marked by the doubling of the bacterial cells. The cell number increases in a logarithmic fashion such that the cell constituent is maintained. The log phase continues until there is depletion of nutrients in the setup. The stage also comes to a stop if toxic substances start to accumulate, resulting in a slower growth rate. The cells are the healthiest at this stage and researchers prefer to use bacteria from this stage for their experimental processes. Plotting this phase on the bacterial growth curve gives a straight line. Upon calculation of the slope of this line, the specific growth rate of the organism is obtained. It is the measure of divisions per cell per unit of time. Stationary Phase In the stationary phase, the rate of growth of the cells becomes equal to its rate of death. The rate of growth of the bacterial cells is limited by the accumulation of toxic compounds and also depletion of nutrients in the media. The cell population remains constant at this stage. Plotting this phase on the graph gives a smooth horizontal linear line. Death Phase This is the last phase of the bacterial growth. At this stage, the rate of death is greater than the rate of formation of new cells. Lack of nutrients, physical conditions or other injuries to the cell leads to death of the cells. This sums up all about the bacterial growth curve.