Probability Distribution in Statistics

Probability Distribution in Statistics If you spend much time at all dealing with statistics, pretty soon you run into the phrase â€Å"probability distribution.† It is here that we really get to see how much the areas of probability and statistics overlap. Although this may sound like something technical, the phrase probability distribution is really just a way to talk about organizing a list of probabilities. A probability distribution is a function or rule that assigns probabilities to each value of a random variable. The distribution may in some cases be listed. In other cases, it is presented as a graph. Example Suppose that we roll two dice and then record the sum of the dice. Sums anywhere from two to 12 are possible. Each sum has a particular probability of occurring. We can simply list these as follows: The sum of 2 has a probability of 1/36The sum of 3 has a probability of 2/36The sum of 4 has a probability of 3/36The sum of 5 has a probability of 4/36The sum of 6 has a probability of 5/36The sum of 7 has a probability of 6/36The sum of 8 has a probability of 5/36The sum of 9 has a probability of 4/36The sum of 10 has a probability of 3/36The sum of 11 has a probability of 2/36The sum of 12 has a probability of 1/36 This list is a probability distribution for the probability experiment of rolling two dice. We can also consider the above as a probability distribution of the random variable defined by looking at the sum of the two dice. Graph A probability distribution can be graphed, and sometimes this helps to show us features of the distribution that were not apparent from just reading the list of probabilities. The random variable is plotted along the x-axis, and the corresponding probability is plotted along the y-axis. For a discrete random variable, we will have a histogram. For a continuous random variable, we will have the inside of a smooth curve. The rules of probability are still in effect, and they manifest themselves in a few ways. Since probabilities are greater than or equal to zero, the graph of a probability distribution must have y-coordinates that are nonnegative. Another feature of probabilities, namely that one is the maximum that the probability of an event can be, shows up in another way. Area Probability The graph of a probability distribution is constructed in such a way that areas represent probabilities. For a discrete probability distribution, we are really just calculating the areas of rectangles. In the graph above, the areas of the three bars corresponding to four, five and six correspond to the probability that the sum of our dice is four, five or six. The areas of all of the bars add up to a total of one. In the standard normal distribution or bell curve, we have a similar situation. The area under the curve between two z values corresponds to the probability that our variable falls between those two values. For example, the area under the bell curve for -1 z. Important Distributions There are literally infinitely many probability distributions. A list of some of the more important distributions follows: Binomial distribution – Gives the number of successes for a series of independent experiments with two outcomesChi-square distribution – For use of determining how close observed quantities fit a proposed modelF-distribution – Used in the analysis of variance (ANOVA)Normal distribution – Called the bell curve and is found throughout statistics.Student’s t distribution – For use with small sample sizes from a normal distribution

Molarity Definition as Used in Chemistry

Molarity Definition as Used in Chemistry In chemistry, molarity is a  concentration unit, defined to be the number of moles of solute divided by the number of liters of solution. Units of Molarity Molarity is expressed in units of moles per liter (mol/L). Its such a common unit, it has its own symbol, which is a capital letter M. A solution that has the concentration 5 mol/L would be called a 5 M solution or said to have a concentration value of 5 molar. Molarity Examples There are 6 moles of HCl in one liter of 6 molar HCl or 6 M HCl.There are 0.05 moles of NaCl in 500 ml of a 0.1 M NaCl solution. (The calculation of moles of ions depends on their solubility.)There are 0.1 moles of Na ions in one liter of a 0.1 M NaCl solution (aqueous). Example Problem Express the concentration of a solution of 1.2 grams of KCl in 250 ml of water. In order to solve the problem, you need to convert the values into the units of molarity, which are moles and liters. Start by converting grams of potassium chloride (KCl) into moles. To do this, look up the atomic masses of the elements on the periodic table. The atomic mass is the mass in grams of 1 mole of atoms. mass of K 39,10 g/molmass of Cl 35.45 g/mol So, the mass of one mole of KCl is: mass of KCl mass of K mass of Clmass of KCl 39.10 g 35.45 gmass of KCl 74.55 g/mol You have 1.2 grams of KCl, so you need to find how many moles that is: moles KCl (1.2 g KCl)(1 mol/74.55 g)moles KCl 0.0161 mol Now, you know how many moles of solute are present. Next, you need to convert the volume of solvent (water) from ml to L. Remember, there are 1000 milliliters in 1 liter: liters of water (250 ml)(1 L/1000 ml)liters of water 0.25 L Finally, youre ready to determine molarity. Simply express the concentration of KCl in water in terms of moles solute (KCl) per liters of solute (water): molarity of solution mol KC/L watermolarity 0.0161 mol KCl/0.25 L watermolarity of the solution 0.0644 M (calculator) Since you were given mass and volume using 2 significant figures, you should report molarity in 2 sig figs also: molarity of KCl solution 0.064 M Advantages and Disadvantages of Using Molarity There are two big advantages of using molarity to express concentration. The first advantage is that its easy and convenient to use because the solute may be measured in grams, converted into moles, and mixed with a volume. The second advantage is that the sum of the molar concentrations is the total molar concentration. This permits calculations of density and ionic strength. The big disadvantage of molarity is that it changes according to temperature. This is because the volume of a liquid is affected by temperature. If measurements are all performed at a single temperature (e.g., room temperature), this is not a problem. However, its good practice to report the temperature when citing a molarity value. When making a solution, keep in mind, molarity will change slightly if you use a hot or cold solvent, yet store the final solution at a different temperature.