# Mean and variance of a random variable

03/01/2021

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## Outline

• The following topics will be covered in this lecture:

• Mean, median and mode
• Weighted mean
• Mean of a frequency distribution
• Mean of a probability distribution
• Standard deviation
• Variance

## Motivation

• Our goal in this course is to use statistics from a small, representative sample to say something general about the larger, unobservable population or phenomena.

• Recall, the measures of the population are what we referred to as parameters.

• Parameters are generally unknown and unknowable.

• For example, the mean age of every adult living in the United States is a parameter for the adult population of the USA.
• We cannot possibly know this value exactly as there are people who cannot be surveyed and / or don't have accurate records.
• If we have a representative sample we can compute the sample mean.
• The sample mean will almost surely not equal population mean, due to the natural variation (sampling error) that occurs in any given sample.
• However, if we have a good probabilistic model for the ages of adults, we can use the sample statistic to estimate the general, unknown population parameter.
• Random variables and probability distributions give us the model for estimating population parameters.

• Note: we can only “find” the parameters exactly in very simple examples like games of chance.

• Generally, we will have to be satisfied with estimates of the parameters that are uncertain, but also include measures of “how uncertain”.

## Characteristics of data Courtesy of M. W. Toews CC via Wikimedia Commons.

• In statistics, we try to characterize data and populations by a number of the features that they exhibits.
• For a single variable, the most common measures are:
1. Center: A representative value that indicates where the middle of the data set is located.
2. Spread: A measure of the amount that the data values vary around the center.
• We will now recall some measures of center:
1. mean;
2. median; and
3. mode.
• Each of these usually gives a different view of where the “most central point” of the data lies.

## Mean

• The (arithmetic sample) mean is usually the most important measure of center.

• Suppose we have a sample of $$n$$ total measurements of some random variable $$X$$.

• We will denote these measurements $$x_1, x_2, \cdots, x_n$$
• Then, the (arithmetic sample) mean is defined

$\text{Sample mean} = \overline{x} = \frac{x_1 +x_2 +\cdots + x_n}{n}= \frac{\sum_{i=1}^n x_i}{n}$

• Q: is the sample mean a statistic or a parameter?

• A: the sample mean is computed from a sample and thus is a statistic.
• For this reason, if we took new measurements from a new sample of the population, we could get a different value.
• The random difference between the sample mean and the mean of the true population mean is called sampling error.
• An important property of the sample mean is that it tends to vary less over re-sampling than other statistics.

• That is, it tends to stay close to the same value.
• However, the sample mean is very sensitive to outliers.

• If outliers exist in the data, the mean can be drawn far away from the “main” cluster of data.
• A statistic is called resistant if it doesn't change very much with respect to outlier data.

## Median and mode

• A different notion of center is the middle of the data.
• For a numerical measurement, we can always order the data so that we go from low to high or high to low.
• Median – the median is the middle of the ordered data set.
• If there are an odd number of measurements, the median is defined as the middle value exactly.
• If there are an even number of measurements, we split the data into the lower $$50\%$$ and upper $$50\%$$ of the measurements;
• then we take the median to be the mean of the:
1. largest of the lower $$50\%$$; and
2. smallest of the upper $$50\%$$.
• Another notion of the most “central point” in the data can be the value that is measured most frequently.
• Mode – the mode is the observed value that is most frequent in the data.