Category: statistics

Statistics – Grand Mean ”; Previous Next When sample sizes are equal, in other words, there could be five values in each sample, or n values in each sample. The grand mean is the same as the mean of sample means. Formula ${X_{GM} = frac{sum x}{N}}$ Where − ${N}$ = Total number of sets. ${sum x}$ = sum of the mean of all sets. Example Problem Statement: Determine the mean of each group or set”s samples. Use the following data as a sample to determine the mean and grand mean. Jackson 1 6 7 10 4 Thomas 5 2 8 14 6 Garrard 8 2 9 12 7 Solution: Step 1: Compute all means $ {M_1 = frac{1+6+7+10+4}{5} = frac{28}{5} = 5.6 \[7pt] , M_2 = frac{5+2+8+14+6}{5} = frac{35}{5} = 7 \[7pt] , M_3 = frac{8+2+9+12+7}{5} = frac{38}{5} = 7.6 }$ Step 2: Divide the total by the number of groups to determine the grand mean. In the sample, there are three groups. $ {X_{GM} = frac{5.6+7+7.6}{3} = frac{20.2}{3} \[7pt] , = 6.73 }$ Print Page Previous Next Advertisements ”;

Statistics – Signal to Noise Ratio ”; Previous Next Sign to-commotion proportion (contracted SNR) is a measure utilized as a part of science and designing that analyzes the level of a coveted sign to the level of foundation clamor. It is characterized as the proportion of sign energy to the clamor power, regularly communicated in decibels. A proportion higher than 1:1 (more prominent than 0 dB) shows more flag than clamor. While SNR is regularly cited for electrical signs, it can be connected to any type of sign, (for example, isotope levels in an ice center or biochemical motioning between cells). Signal-to-noise ratio is defined as the ratio of the power of a signal (meaningful information) and the power of background noise (unwanted signal): ${SNR = frac{P_{signal}}{P_{noise}}}$ If the variance of the signal and noise are known, and the signal is zero: ${SNR = frac{sigma^2_{signal}}{sigma^2_{noise}}}$ If the signal and the noise are measured across the same impedance, then the SNR can be obtained by calculating the square of the amplitude ratio: ${SNR = frac{P_{signal}}{P_{noise}} = {(frac{A_{signal}}{A_{noise}})}^2} $ Where A is root mean square (RMS) amplitude (for example, RMS voltage). Decibels Because many signals have a very wide dynamic range, signals are often expressed using the logarithmic decibel scale. Based upon the definition of decibel, signal and noise may be expressed in decibels (dB) as ${P_{signal,dB} = 10log_{10}(P_{signal})} $ and ${P_{noise,dB} = 10log_{10}(P_{noise})} $ In a similar manner, SNR may be expressed in decibels as ${SNR_{dB} = 10log_{10}(SNR)} $ Using the definition of SNR ${SNR_{dB} = 10log_{10}(frac{P_{signal}}{P_{noise}})} $ Using the quotient rule for logarithms ${10log_{10}(frac{P_{signal}}{P_{noise}}) = 10log_{10}(P_{signal}) – 10log_{10}(P_{noise})} $ Substituting the definitions of SNR, signal, and noise in decibels into the above equation results in an important formula for calculating the signal to noise ratio in decibels, when the signal and noise are also in decibels: ${SNR_{dB} = P_{signal,dB} – P_{noise,dB}} $ In the above formula, P is measured in units of power, such as Watts or mill watts, and signal-to-noise ratio is a pure number. However, when the signal and noise are measured in Volts or Amperes, which are measures of amplitudes, they must be squared to be proportionate to power as shown below: ${SNR_{dB} = 10log_{10}[{(frac{A_{signal}}{A_{noise}})}^2] \[7pt] = 20log_{10}(frac{A_{signal}}{A_{noise}}) \[7pt] = A_{signal,dB} – A_{noise,dB}} $ Example Problem Statement: Compute the SNR of a 2.5 kHz sinusoid sampled at 48 kHz. Add white noise with standard deviation 0.001. Set the random number generator to the default settings for reproducible results. Solution: ${ F_i = 2500; F_s = 48e3; N = 1024; \[7pt] x = sin(2 times pi times frac{F_i}{F_s} times (1:N)) + 0.001 times randn(1,N); \[7pt] SNR = snr(x,Fs) \[7pt] SNR = 57.7103}$ Print Page Previous Next Advertisements ”;

Statistics – Hypothesis testing ”; Previous Next A statistical hypothesis is an assumption about a population which may or may not be true. Hypothesis testing is a set of formal procedures used by statisticians to either accept or reject statistical hypotheses. Statistical hypotheses are of two types: Null hypothesis, ${H_0}$ – represents a hypothesis of chance basis. Alternative hypothesis, ${H_a}$ – represents a hypothesis of observations which are influenced by some non-random cause. Example suppose we wanted to check whether a coin was fair and balanced. A null hypothesis might say, that half flips will be of head and half will of tails whereas alternative hypothesis might say that flips of head and tail may be very different. $ H_0: P = 0.5 \[7pt] H_a: P ne 0.5 $ For example if we flipped the coin 50 times, in which 40 Heads and 10 Tails results. Using result, we need to reject the null hypothesis and would conclude, based on the evidence, that the coin was probably not fair and balanced. Hypothesis Tests Following formal process is used by statistican to determine whether to reject a null hypothesis, based on sample data. This process is called hypothesis testing and is consists of following four steps: State the hypotheses – This step involves stating both null and alternative hypotheses. The hypotheses should be stated in such a way that they are mutually exclusive. If one is true then other must be false. Formulate an analysis plan – The analysis plan is to describe how to use the sample data to evaluate the null hypothesis. The evaluation process focuses around a single test statistic. Analyze sample data – Find the value of the test statistic (using properties like mean score, proportion, t statistic, z-score, etc.) stated in the analysis plan. Interpret results – Apply the decisions stated in the analysis plan. If the value of the test statistic is very unlikely based on the null hypothesis, then reject the null hypothesis. Print Page Previous Next Advertisements ”;

Statistics – Harmonic Mean ”; Previous Next What is Harmonic Mean? Harmonic Mean is also a mathematical average but is limited in its application. It is generally used to find average of variables that are expressed as a ratio of two different measuring units e. g. speed is measured in km/hr or miles/sec etc. Weighted Harmonic Mean Formula $H.M. = frac{W}{sum (frac{W}{X})}$ Where − ${H.M.}$ = Harmonic Mean ${W}$ = Weight ${X}$ = Variable value Example Problem Statement: Find the weighted H.M. of the items 4, 7,12,19,25 with weights 1, 2,1,1,1 respectively. Solution: ${X}$ ${W}$ $frac{W}{X}$ 4 1 0.2500 7 2 0.2857 12 1 0.0833 19 1 0.0526 25 1 0.0400   $sum W$ $sum frac{W}{X}$= 0.7116 Based on the above mentioned formula, Harmonic Mean $G.M.$ will be: $H.M. = frac{W}{sum (frac{W}{X})} \[7pt] , = frac{6}{0.7116} \[7pt] , = 8.4317 $ ∴ Weighted H.M = 8.4317 We”re going to discuss methods to compute the Harmonic Mean for three types of series: Individual Data Series Discrete Data Series Continuous Data Series Individual Data Series When data is given on individual basis. Following is an example of individual series: Items 5 10 20 30 40 50 60 70 Discrete Data Series When data is given alongwith their frequencies. Following is an example of discrete series: Items 5 10 20 30 40 50 60 70 Frequency 2 5 1 3 12 0 5 7 Continuous Data Series When data is given based on ranges alongwith their frequencies. Following is an example of continous series: Items 0-5 5-10 10-20 20-30 30-40 Frequency 2 5 1 3 12 Print Page Previous Next Advertisements ”;

Statistics – Individual Series Arithmetic Mode ”; Previous Next When data is given on individual basis. Following is an example of individual series − Items 5 10 20 30 40 50 60 70 In case of individual items, the number of times each value occurs is counted and the value which is repeated maximum number of times is the modal value. Example Problem Statement − Calculate Arithmetic Mode for the following individual data − Items 14 36 45 36 105 36 Solution − The Arithmetic Mode of the given numbers is 36 as it is repeated maximum number of times,3. Calculator Print Page Previous Next Advertisements ”;

Statistics – Negative Binomial Distribution ”; Previous Next Negative binomial distribution is a probability distribution of number of occurences of successes and failures in a sequence of independent trails before a specific number of success occurs. Following are the key points to be noted about a negative binomial experiment. The experiment should be of x repeated trials. Each trail have two possible outcome, one for success, another for failure. Probability of success is same on every trial. Output of one trial is independent of output of another trail. Experiment should be carried out until r successes are observed, where r is mentioned beforehand. Negative binomial distribution probability can be computed using following: Formula ${ f(x; r, P) = ^{x-1}C_{r-1} times P^r times (1-P)^{x-r} }$ Where − ${x}$ = Total number of trials. ${r}$ = Number of occurences of success. ${P}$ = Probability of success on each occurence. ${1-P}$ = Probability of failure on each occurence. ${f(x; r, P)}$ = Negative binomial probability, the probability that an x-trial negative binomial experiment results in the rth success on the xth trial, when the probability of success on each trial is P. ${^{n}C_{r}}$ = Combination of n items taken r at a time. Example Robert is a football player. His success rate of goal hitting is 70%. What is the probability that Robert hits his third goal on his fifth attempt? Solution: Here probability of success, P is 0.70. Number of trials, x is 5 and number of successes, r is 3. Using negative binomial distribution formula, let”s compute the probability of hitting third goal in fifth attempt. ${ f(x; r, P) = ^{x-1}C_{r-1} times P^r times (1-P)^{x-r} \[7pt] implies f(5; 3, 0.7) = ^4C_2 times 0.7^3 times 0.3^2 \[7pt] , = 6 times 0.343 times 0.09 \[7pt] , = 0.18522 }$ Thus probability of hitting third goal in fifth attempt is $ { 0.18522 }$. Print Page Previous Next Advertisements ”;

Statistics – Histograms ”; Previous Next A histogram is a graphical representation of the distribution of numerical data. It is an estimate of the probability distribution of a continuous variable (quantitative variable). Problem Statement: Every month one measure the amount of weight one”s dog has picked up and get these outcomes: 0.5 0.5 0.3 -0.2 1.6 0 0.1 0.1 0.6 0.4 Draw the histogram demonstrating how much is that dog developing. Solution: monthly development vary from -0.2 (the fox lost weight that month) to 1.6. Putting them in order from lowest to highest weight gain. -0.2 0 0.1 0.1 0.3 0.4 0.5 0.5 0.6 1.6 We decide to put the results into groups of 0.5: The -0.5 to just below 0 range. The 0 to just below 0.5 range, etc. And here is the result: There are no values from 1 to just below 1.5, but we still show the space. Print Page Previous Next Advertisements ”;

Statistics – Individual Series Arithmetic Median ”; Previous Next When data is given on individual basis. Following is an example of individual series − Items 5 10 20 30 40 50 60 70 In case of a group having even number of distribution, Arithmetic Median is found out by taking out the Arithmetic Mean of two middle values after arranging the numbers in ascending order. Formula Median = Value of ($frac{N+1}{2})^{th} item$. Where − ${N}$ = Number of observations Example Problem Statement − Let”s calculate Arithmetic Median for the following individual data − Items 14 36 45 70 105 145 Solution − Based on the above mentioned formula, Arithmetic Median M will be − $M = Value of (frac{N+1}{2})^{th} item. \[7pt] , = Value of (frac{6+1}{2})^{th} item. \[7pt] , = Value of 3.5^{th} item. \[7pt] , = Value of (frac{3^{rd} item + 4^{th} item}{2})\[7pt] , = (frac{45 + 70}{2}) , = {57.5}$ The Arithmetic Median of the given numbers is 57.5. In case of a group having odd number of distribution, Arithmetic Median is the middle number after arranging the numbers in ascending order. Example Let”s calculate Arithmetic Median for the following individual data − Items 14 36 45 70 105 Given numbers are 5, an odd number thus middle number is the Arithmetic Median. ∴ The Arithmetic Median of the given numbers is 45. Calculator Print Page Previous Next Advertisements ”;

Statistics – Inverse Gamma Distribution ”; Previous Next Inverse Gamma Distribution is a reciprocal of gamma probability density function with positive shape parameters $ {alpha, beta } $ and location parameter $ { mu } $. $ {alpha } $ controls the height. Higher the $ {alpha } $, taller is the probability density function (PDF). $ {beta } $ controls the speed. It is defined by following formula. Formula ${ f(x) = frac{x^{-(alpha+1)}e^{frac{-1}{beta x}}}{ Gamma(alpha) beta^alpha} \[7pt] , where x gt 0 }$ Where − ${alpha}$ = positive shape parameter. ${beta}$ = positive shape parameter. ${x}$ = random variable. Following diagram shows the probability density function with different parameter combinations. Print Page Previous Next Advertisements ”;

Statistics – Relative Standard Deviation ”; Previous Next In probability theory and statistics, the coefficient of variation (CV), also known as relative standard deviation (RSD), is a standardized measure of dispersion of a probability distribution or frequency distribution. Relative Standard Deviation, RSD is defined and given by the following probability function: Formula ${100 times frac{s}{bar x}}$ Where − ${s}$ = the sample standard deviation ${bar x}$ = sample mean Example Problem Statement: Find the RSD for the following set of numbers: 49, 51.3, 52.7, 55.8 and the standard deviation are 2.8437065. Solution: Step 1 – Standard deviation of sample: 2.8437065 (or 2.84 rounded to 2 decimal places). Step 2 – Multiply Step 1 by 100. Set this number aside for a moment. ${2.84 times 100 = 284}$ Step 3 – Find the sample mean, ${bar x}$. The sample mean is: ${frac{(49 + 51.3 + 52.7 + 55.8)}{4} = frac{208.8}{4} = 52.2.}$ Step 4Divide Step 2 by the absolute value of Step 3. ${frac{284}{|52.2|} = 5.44.}$ The RSD is: ${52.2 pm 5.4}$% Note that the RSD is expressed as a percentage. Print Page Previous Next Advertisements ”;