A PDF is simply the derivative of a CDF. Thus a PDF is also a function of a random variable, x, and its magnitude will be some indication of the relative likelihood of measuring a particular value. Furthermore and by definition, the area under the curve of a PDF(x) between -∞ and x equals its CDF(x).

To select the correct probability distribution:

1. Look at the variable in question.
2. Review the descriptions of the probability distributions.
3. Select the distribution that characterizes this variable.
4. If historical data are available, use distribution fitting to select the distribution that best describes your data.

A CDF function, such as F(x), is the integral of the PDF f(x) up to x. That is, the probability of getting a value x or smaller P(Y <= x) = F(x). So if you want to find the probability of rain between 1.9 < Y < 2.1 you can use F(2.1) – F(1.9), which is equal to integrating f(x) from x = 1.9 to 2.1.

The cumulative distribution function (CDF) calculates the cumulative probability for a given x-value. Use the CDF to determine the probability that a random observation that is taken from the population will be less than or equal to a certain value.

The CDF of the standard normal distribution is denoted by the Φ function: Φ(x)=P(Z≤x)=1√2π∫x−∞exp{−u22}du. As we will see in a moment, the CDF of any normal random variable can be written in terms of the Φ function, so the Φ function is widely used in probability.

In a normal distribution the mean is zero and the standard deviation is 1. It has zero skew and a kurtosis of 3. Normal distributions are symmetrical, but not all symmetrical distributions are normal.

The normal distribution is often called the bell curve because the graph of its probability density looks like a bell. It is also known as called Gaussian distribution, after the German mathematician Carl Gauss who first described it.

One reason the normal distribution is important is that many psychological and educational variables are distributed approximately normally. Measures of reading ability, introversion, job satisfaction, and memory are among the many psychological variables approximately normally distributed.

SOLUTION: The middle 99.7% of data in a normal distribution is the range from µ – 3σ to µ + 3σ.

So the mean and median of a normal distribution are the same. Since a normal distribution is also symmetric about its highest peak, the mode (as well as the mean and median) are all equal in a normal distribution.

Mean is the average of all the values. Median is the middle value, dividing the number of data into 2 halves. In other words, 50% of the observations is below the median and 50% of the observations are above the median. Mode is the most common value among the given observations.

To summarize, generally if the distribution of data is skewed to the left, the mean is less than the median, which is often less than the mode. If the distribution of data is skewed to the right, the mode is often less than the median, which is less than the mean.

Mean is the most frequently used measure of central tendency and generally considered the best measure of it. Median is the preferred measure of central tendency when: There are a few extreme scores in the distribution of the data.

mean

However, in this situation, the mean is widely preferred as the best measure of central tendency because it is the measure that includes all the values in the data set for its calculation, and any change in any of the scores will affect the value of the mean.

The median is usually preferred to other measures of central tendency when your data set is skewed (i.e., forms a skewed distribution) or you are dealing with ordinal data. However, the mode can also be appropriate in these situations, but is not as commonly used as the median.

For negatively skewed distributions, the mean will always be the lowest estimate of central tendency and the mode will be the highest estimate of central tendency. In any skewed distribution (i.e., positive or negative) the median will always fall in-between the mean and the mode.

Solution. Standard deviation is not a measure of central tendency.

Unlike the mean, the median value doesn’t depend on all the values in the dataset. Consequently, when some of the values are more extreme, the effect on the median is smaller. When you have a skewed distribution, the median is a better measure of central tendency than the mean.

The “mean” is the “average” you’re used to, where you add up all the numbers and then divide by the number of numbers. The “median” is the “middle” value in the list of numbers.

Median. The median is the midpoint of the data set. This midpoint value is the point at which half the observations are above the value and half the observations are below the value. The median is determined by ranking the observations and finding the observation that are at the number [N + 1] / 2 in the ranked order.

WHAT CAN THE MEDIAN TELL YOU? The median provides a helpful measure of the centre of a dataset. By comparing the median to the mean, you can get an idea of the distribution of a dataset. When the mean and the median are the same, the dataset is more or less evenly distributed from the lowest to highest values.

One of the basic tenets of statistics that every student learns in about the second week of intro stats is that in a skewed distribution, the mean is closer to the tail in a skewed distribution. So in a right skewed distribution (the tail points right on the number line), the mean is higher than the median.

The median is the value in the center of the data. Half of the values are less than the median and half of the values are more than the median. It is probably the best measure of center to use in a skewed distribution. Once the depth of the median is found, the median is the value in that position.

The median can be used to determine an approximate average, or mean, but is not to be confused with the actual mean. If there is an odd amount of numbers, the median value is the number that is in the middle, with the same amount of numbers below and above.

Median

1. Arrange your numbers in numerical order.
2. Count how many numbers you have.
3. If you have an odd number, divide by 2 and round up to get the position of the median number.
4. If you have an even number, divide by 2. Go to the number in that position and average it with the number in the next higher position to get the median.

The median is different for different types of distribution. For example, the median of 3, 3, 5, 9, 11 is 5. If there is an even number of observations, then there is no single middle value; the median is then usually defined to be the mean of the two middle values: so the median of 3, 5, 7, 9 is (5+7)/2 = 6.

The median of a normal distribution with mean μ and variance σ2 is μ. In fact, for a normal distribution, mean = median = mode. The median of a uniform distribution in the interval [a, b] is (a + b) / 2, which is also the mean.