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From Wikipedia, the free encyclopedia

There are several kinds of means in various branches of mathematics (especially statistics).

For a data set, the arithmetic mean, also called the mathematical expectation or average, is the central value of a discrete set of numbers: specifically, the sum of the values divided by the number of values. The arithmetic mean of a set of numbers x1, x2, ..., xn is typically denoted by , pronounced "x bar". If the data set were based on a series of observations obtained by sampling from a statistical population, the arithmetic mean is the sample mean (denoted ) to distinguish it from the mean of the underlying distribution, the population mean (denoted or ).[1] Pronounced "mew" /'mjuː/.

In probability and statistics, the population mean, or expected value, are a measure of the central tendency either of a probability distribution or of the random variable characterized by that distribution.[2] In the case of a discrete probability distribution of a random variable X, the mean is equal to the sum over every possible value weighted by the probability of that value; that is, it is computed by taking the product of each possible value x of X and its probability p(x), and then adding all these products together, giving .[3] An analogous formula applies to the case of a continuous probability distribution. Not every probability distribution has a defined mean; see the Cauchy distribution for an example. Moreover, for some distributions the mean is infinite.

For a finite population, the population mean of a property is equal to the arithmetic mean of the given property while considering every member of the population. For example, the population mean height is equal to the sum of the heights of every individual divided by the total number of individuals. The sample mean may differ from the population mean, especially for small samples. The law of large numbers dictates that the larger the size of the sample, the more likely it is that the sample mean will be close to the population mean.[4]

Outside probability and statistics, a wide range of other notions of "mean" are often used in geometry and analysis; examples are given below.

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  • ✪ Math Antics - Mean, Median and Mode
  • ✪ Statistics (सांख्यिकी) - Mean, Median & Mode (माध्य, मध्यिका और बहुलक)
  • ✪ Find the mean or the average
  • ✪ Mean using Direct Method & Shortcut Method
  • ✪ Math Antics - Order Of Operations

Transcription

Hi, this is Rob. Welcome to Math Antics! In this lesson, we’re gonna learn about three important math concepts called the Mean, the Median and the Mode. Math often deals with data sets, and data sets are often just collections (or groups) of numbers. These numbers may be the results of scientific measurements or surveys or other data collection methods. For example, you might record the ages of each member of you family into a data set. Or you might measure the weight of each of your pets and list them in a data set. Those data sets are fairly small and easy to understand. But you could have much bigger data sets. A really big data set might contain the cost of every item in a store, or the top speed of every land mammal, or the brightness of all the stars in our galaxy! Those data sets would contain a lot of different numbers! And if you had to look at a big data set all at one time… it would be pretty hard to make sense of it or say much about it besides “well that’s a lot of numbers”! But that’s where Mean, Median and Mode can really help us out. They’re three different properties of data sets that can give us useful, easy to understand information about a data set so that we can see the big picture and understand what the data means about the world we live in. That sounds pretty useful, huh? So let’s learn what each property really is and find out how to calculate them for any particular data set. Let’s start with the Mean. You may not have ever heard of something called “the mean” before, but I’ll bet you’ve heard of “the average”. If so, then I’ve got good news! Mean means average! “Mean” and “average” are just two different terms for the exact same property of a data set. The mean (or average) is an extremely useful property. To understand what it is, let’s look at a simple data set that contains 5 numbers. As a visual aid, let’s also represent those numbers with stacks of blocks who’s heights correspond to their values: one, eight, three, two, six Right now, since each of the 5 numbers is different, the stacks of blocks are all different heights. But what if we rearrange the blocks with the goal of making the stacks the same height? In other words, if each stack could have the exact same amount, what would that amount be? Well, with a bit of trial and error, you’ll see that we have enough blocks for each stack to have a total of 4. That means that the Mean (or average) for our original data set would be 4. Some of the numbers are greater than 4 and some are less, but if the amounts could all be made the same, they would all become 4. So that’s the concept of Mean; it’s the value you’d get if you could smooth out or flatten all of the different data values into one consistent value. But, is there a way we can use math to calculate the mean of a data set? After all, it would be very inconvenient if we always had to use stacks of blocks to do it! There’s got to be an easier way!! [crash] To learn the mathematical procedure for calculating the Mean, lets start with blocks again. But this time, instead of using trial and error, let’s use a more systematic way to make the stacks all the same height. This way involves a clever combination of addition and division. We know that we want to end up with 5 stacks that all have the same number of blocks, right? So first, let’s add up all of the numbers, which is like putting all of the blocks we have into one big stack. Adding up all of the numbers (or counting all the blocks) shows us that we have a total of 20. Next, we divide that number (or stack) into 5 equal parts. Since the stack has a total of 20 blocks, dividing it into 5 equal stacks means that we’ll have 4 in each, since 20 divided by 5 equals 4. So that’s the math procedure you use to find the mean of a data set. It’s just two simple steps. First, you add up all the numbers in the set. And then you divide the total you get by how many numbers you added up. The answer you get is the Mean of the data set. Let’s use that procedure to find the mean age of the members of this fine looking family here. If we add them all up using a calculator (or by hand if you’d like) the total of the ages is 222 years. But then, we need to divide that total by the number of ages we added which is 6. 222 divided by 6 is 37. So that’s the mean age of all the members in this family. Alright, that’s the Mean. Now what about the Median? The Median is the middle of a data set. It’s the number that splits the data set into two equally sized group or halves. One half contains members that are greater than or equal to the Median, and the other half contains members that are less than or equal to the Median. Sometimes finding the Median of a data set is easy, and sometimes it’s hard. That’s because finding the middle value of a data set requires that its members be in order from the least to the greatest (or vice versa). And if the data set has a lot of numbers, it might take a lot of work to put them in the right order if they aren’t already that way. So to make things easier, let’s start with a really basic data set that isn’t in order. It’s pretty easy to see that we can put this data set in order from the least to the greatest value just by switching the 2 and the 1. There, now we have the data set {1, 2, 3} and finding the Median (or middle) of this data set is easy! It’s just 2 because the 2 is located exactly in the middle. That almost seems too easy, doesn’t it? But don’t worry… it gets harder! But before we try a harder problem, I want to point out that sometimes the Mean and the Median of a data set are the same number, and sometimes they’re not. In the case of our simple data set {1, 2, 3}, the Median is 2 and the Mean is also 2, as you can see if we rearrange the amounts or follow the procedure we learned to calculate the Mean. But what about the first data set that we found the mean of? We determined than the Mean of this data set is 4. But what about the Median? Well, the Median is the middle, and since this data set is already in order from least to greatest, it’s easy to see that the 3 is located in the middle since it splits the other members into two equal groups. So for this data set, the Mean is 4 but the Median is 3. So to find the Median of a set of numbers, first you need to make sure that all the numbers are in order and then you can identify the member that’s exactly in the middle by making sure there’s an equal number of members on either side of it. Okay, ...so far so good. But some of you may be wondering, “What if a data set doesn’t have an obvious middle member?” All of the sets we’ve found the Median of so far have an odd number of members. But, what if it has an even number of members? …like the data set {1, 2, 3, 4} There isn’t a member in the middle that splits the set into two equally sized groups. If that’s the case, we can actually use what we learned about the Mean to help us out. If the data set has an even number of members, then to find the Median, we need to take the middle TWO numbers and calculate the Mean (or average) of those two. By doing that, we’re basically figuring out what number WOULD be exactly half way between the two middle numbers, and that number will be our Median. For example, in the set {1, 2, 3, 4} we need to take the middle TWO numbers (2 and 3) and find the Mean of those numbers. We can do that by adding 2 and 3 and then dividing by 2. 2 plus 3 equals 5 and 5 divided by 2 is 2.5 So the Median of the data set is 2.5 Even though the number 2.5 isn’t actually a member of the data set, it’s the Median because it represents the middle of the data set and it splits the members into two equally sized groups. Okay, so now you know the difference between Mean and Median. But what about the Mode of a data set? What in the world does that mean? Well, “Mode” is just a technical word for the value in a data set that occurs most often. In the data sets we’ve seen so far, there hasn’t even been a Mode because none of the data values were ever repeated. But what if you had this data set? This set has 6 members, but some of the value are repeated. If we rearrange them, you can see that there’s one ‘1’, two ‘2’s and three ‘3’s The Mode of this data set is the value that occurs most often (or most frequently) so that would be 3 since there’s three ‘3’s. Now don’t get confused just because the number 3 was repeater 3 times. The mode is the number that’s repeated most often, NOT how many times it was repeated. As I mentioned, if each member in a data set occurs only once, it had no mode, but it’s also possible for a data set to have more than one mode. Here’s an example of a data set like that: In this set, the number 7 is repeated twice but so is the number 15. That means they tie for the title of Mode. This set has two modes: 7 and 15. Okay, so now that you know what the Mean, Median and Mode of a data set are. Let’s put all that new information to use on one final real-world example. Suppose there’s this guy who makes and sells custom electric guitars. Here’s a table showing how many guitars he sold during each month of the year. Let’s find the Mean, Median and Mode of this data set. First, to find the Mean we need to add up the number of guitars sold in each month. You can do the addition by hand or you can use a calculator if you want to. Either way, be careful since that’s a lot of numbers to add up and we don’t want to make a mistake. The answer I get is 108. So that’s the total he sold for the whole year, but to get the Mean sold each month, we need to divide that total by the number of months which is 12. 108 divided by 12 is 9, so the Mean (or average) is 9. Next, to find the Median of the data set, we’re going to have to rearrange the 12 data points in order from smallest to largest so we can figure out what the middle value is. There, that’s better. Since there’s an even number of members in this set, we can’t just choose the middle number, so we’re going to have to pick the middle two numbers and then find the Mean of them. 9 and 10 are in the middle since there’s an equal number of data values on either side of them. So we need to take the Mean of 9 and 10. That’s easy, 9 plus 10 equals 19 and then 19 divided by 2 is 9.5 So, the Median number of guitars sold is 9.5. That means that in half of the months, he sold more than 9.5, and in half of the months, he sold less than 9.5. Last of all, let’s identify the Mode of this data set (if there is one). We let’s see… there’s two ‘8’s in the data set… Oh… but there’s three ’10’s. That looks like the most frequent number, so 10 is the Mode of this data set. It’s the result that occurred most often. Alright, so that’s the basics of Mean, Median, and Mode. They are three really useful properties of data sets and now you know how to find them. But sometimes, the hardest part about Mean, Median and Mode is just remembering which is which. So remember that “Mean means average”, Median is in the middle, and Mode starts with ‘M’ ‘O’ which can remind you that it’s the number that occurs “Most Often”. Remember, to get good at math, you need to do more than just watch videos about it. You need to Practice! So be sure to try finding the Mean, Median and Mode on your own. As always, thank for watching Math Antics, and I’ll see ya next time. Learn more at www.mathantics.com

Contents

Types of mean

Pythagorean means

Arithmetic mean (AM)

The arithmetic mean (or simply mean) of a sample , usually denoted by , is the sum of the sampled values divided by the number of items in the example

For example, the arithmetic mean of five values: 4, 36, 45, 50, 75 is:

Geometric mean (GM)

The geometric mean is an average that is useful for sets of positive numbers that are interpreted according to their product and not their sum (as is the case with the arithmetic mean); e.g., rates of growth.

For example, the geometric mean of five values: 4, 36, 45, 50, 75 is:

Harmonic mean (HM)

The harmonic mean is an average which is useful for sets of numbers which are defined in relation to some unit, for example speed (distance per unit of time).

For example, the harmonic mean of the five values: 4, 36, 45, 50, 75 is

Relationship between AM, GM, and HM

AM, GM, and HM satisfy these inequalities:

Equality holds if and only if all the elements of the given sample are equal.

Statistical location

Comparison of the arithmetic mean, median and mode of two skewed (log-normal) distributions.
Comparison of the arithmetic mean, median and mode of two skewed (log-normal) distributions.
Geometric visualization of the mode, median and mean of an arbitrary probability density function.[5]
Geometric visualization of the mode, median and mean of an arbitrary probability density function.[5]

In descriptive statistics, the mean may be confused with the median, mode or mid-range, as any of these may be called an "average" (more formally, a measure of central tendency). The mean of a set of observations is the arithmetic average of the values; however, for skewed distributions, the mean is not necessarily the same as the middle value (median), or the most likely value (mode). For example, mean income is typically skewed upwards by a small number of people with very large incomes, so that the majority have an income lower than the mean. By contrast, the median income is the level at which half the population is below and half is above. The mode income is the most likely income and favors the larger number of people with lower incomes. While the median and mode are often more intuitive measures for such skewed data, many skewed distributions are in fact best described by their mean, including the exponential and Poisson distributions.

Mean of a probability distribution

The mean of a probability distribution is the long-run arithmetic average value of a random variable having that distribution. In this context, it is also known as the expected value. For a discrete probability distribution, the mean is given by , where the sum is taken over all possible values of the random variable and is the probability mass function. For a continuous distribution,the mean is , where is the probability density function. In all cases, including those in which the distribution is neither discrete nor continuous, the mean is the Lebesgue integral of the random variable with respect to its probability measure. The mean need not exist or be finite; for some probability distributions the mean is infinite (+∞ or −∞), while others have no mean.

Generalized means

Power mean

The generalized mean, also known as the power mean or Hölder mean, is an abstraction of the quadratic, arithmetic, geometric and harmonic means. It is defined for a set of n positive numbers xi by

By choosing different values for the parameter m, the following types of means are obtained:

maximum of
quadratic mean
arithmetic mean
geometric mean
harmonic mean
minimum of

ƒ-mean

This can be generalized further as the generalized ƒ-mean

and again a suitable choice of an invertible ƒ will give

arithmetic mean,
harmonic mean,
power mean,
geometric mean.

Weighted arithmetic mean

The weighted arithmetic mean (or weighted average) is used if one wants to combine average values from samples of the same population with different sample sizes:

The weights represent the sizes of the different samples. In other applications, they represent a measure for the reliability of the influence upon the mean by the respective values.

Truncated mean

Sometimes a set of numbers might contain outliers, i.e., data values which are much lower or much higher than the others. Often, outliers are erroneous data caused by artifacts. In this case, one can use a truncated mean. It involves discarding given parts of the data at the top or the bottom end, typically an equal amount at each end and then taking the arithmetic mean of the remaining data. The number of values removed is indicated as a percentage of the total number of values.

Interquartile mean

The interquartile mean is a specific example of a truncated mean. It is simply the arithmetic mean after removing the lowest and the highest quarter of values.

assuming the values have been ordered, so is simply a specific example of a weighted mean for a specific set of weights.

Mean of a function

In some circumstances mathematicians may calculate a mean of an infinite (even an uncountable) set of values. This can happen when calculating the mean value of a function . Intuitively this can be thought of as calculating the area under a section of a curve and then dividing by the length of that section. This can be done crudely by counting squares on graph paper or more precisely by integration. The integration formula is written as:

Care must be taken to make sure that the integral converges. But the mean may be finite even if the function itself tends to infinity at some points.

Mean of angles and cyclical quantities

Angles, times of day, and other cyclical quantities require modular arithmetic to add and otherwise combine numbers. In all these situations, there will not be a unique mean. For example, the times an hour before and after midnight are equidistant to both midnight and noon. It is also possible that no mean exists. Consider a color wheel -- there is no mean to the set of all colors. In these situations, you must decide which mean is most useful. You can do this by adjusting the values before averaging, or by using a specialized approach for the mean of circular quantities.

Fréchet mean

The Fréchet mean gives a manner for determining the "center" of a mass distribution on a surface or, more generally, Riemannian manifold. Unlike many other means, the Fréchet mean is defined on a space whose elements cannot necessarily be added together or multiplied by scalars. It is sometimes also known as the Karcher mean (named after Hermann Karcher).

Other means

Distribution of the sample mean

The arithmetic mean of a population, or population mean, is often denoted μ. The sample mean (the arithmetic mean of a sample of values drawn from the population) makes a good estimator of the population mean, as its expected value is equal to the population mean (that is, it is an unbiased estimator). The sample mean is a random variable, not a constant, since its calculated value will randomly differ depending on which members of the population are sampled, and consequently it will have its own distribution. For a random sample of n independent observations, the expected value of the sample mean is

and the variance of the sample mean is

If the population is normally distributed, then the sample mean is normally distributed:

If the population is not normally distributed, the sample mean is nonetheless approximately normally distributed if n is large and σ2/n < +∞. This follows from the central limit theorem.

The mean of a list is all of the numbers added together and divided by the amount of numbers

See also

References

  1. ^ Underhill, L.G.; Bradfield d. (1998) Introstat, Juta and Company Ltd. ISBN 0-7021-3838-X p. 181
  2. ^ Feller, William (1950). Introduction to Probability Theory and its Applications, Vol I. Wiley. p. 221. ISBN 0471257087.
  3. ^ Elementary Statistics by Robert R. Johnson and Patricia J. Kuby, p. 279
  4. ^ Schaum's Outline of Theory and Problems of Probability by Seymour Lipschutz and Marc Lipson, p. 141
  5. ^ "AP Statistics Review - Density Curves and the Normal Distributions". Retrieved 16 March 2015.


This page was last edited on 21 May 2019, at 13:50
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