Means
Means can be calculated in a number of different ways. The most common ones are:
Arithmetic Mean (or just Mean) = $\frac{x1+x2+....+xn}{n}$.
Weighted Arithmetic Mean = $\frac{x1.p1+x2.p2+....+xn.pn}{p1+p2+....+pn}$
In the weighted arithmetic mean,
$x1, x2,.....xn$ are Real numbers, and
$p1, p2,.....pn$ are the weights of each of the data points. The weights can be frequencies or any other factor that indicates how much more some data points contribute to the mean. When all the weights are the same, the weighted arithmet mean is equal to the arithmetic mean.
Geometric Mean = $\sqrt[n]{x1.x2.....xn}$
Harmonic Mean = $1/(\frac{\frac{1}{x1}+\frac{1}{x2}+....+\frac{1}{xn}}{n})$
Quadratic Mean = $\sqrt[n]{\frac{{x1^2+x2^2+...+xn^2}}{n}}$
Let's calculate the means for the set of numbers [1, 3, 5], with weights 0.1, 0.3, and 0.6, respectively.
Arithmetic Mean = $\frac{1+3+5}{3}=3$
Weighted Arithmetic Mean = $\frac{1*0.1+3*0.3+5*0.6}{0.1+0.3+0.6}=4$. Note that if the weights were all the same (2, for example), the weighted arithmetic mean would be 3 (equal to the arithmetic mean).
Geometric Mean = $\sqrt[3]{1.3.5}=\sqrt[3]{15}$
Harmonic Mean = $1/(\frac{\frac{1}{1}+\frac{1}{3}+\frac{1}{5}}{3})=1/(\frac{23}{45})=\frac{45}{23}$
Quadratic Mean = $\sqrt[3]{\frac{{1^2+3^2+5^2}}{3}}=\sqrt[3]{\frac{35}{3}}$
Median
When we sort a list of numbers by either ascending or descending value, the median is the middle number (if the list has an odd number of data), or the arithmetic mean of the two middle numbers (if the sample has an even number of data). For example:
List [1; 4; 5; 7; 8]. The Median is 5.
List [1; 4; 5; 7; 8; 10]. The Median is $\frac{5+7}{2}=6$.
Mode
The Mode of a list of numbers is the value that appears most often. A list can have one, more than one, or no mode at all (if all the numbers appear with the same frequency, the list has no mode).
Examples:
The Mode of the list [1; 4; 4; 5; 7; 8] is 4.
The list [1; 4; 4; 5; 7; 7; 8; 10] has two Modes, 4 and 7.
The list [1; 4; 5; 7; 8] has no Mode.
Dispersion
In statistics, dispersion describes how spread out a distribution is. Some of the most important measures of dispersion are amplitude, mean absolute deviation, variance, and standard deviation.
Amplitude of a set of data is simply the difference between its largest and its smallest values. For example, the amplitude of the set [1; 4; 5; 7; 8] is $8-1=7$
Mean Absolute Deviation (or Mean Absolute Error) is given by the formula:
MAD = $\frac{\sum_{i=1}^{n}|xi-\bar{x}|}{n}$, where $\bar{x}$ is the arithmetic mean of the set of data, and $n$ is the number of elements in the set.
For example, in the set of data [1; 4; 5; 7; 8], where $\bar{x}=5$ and $n=5$, the mean absolute deviation is:
MAD = $\frac{\sum_{i=1}^{5}|xi-5|}{5}$
MAD = $\frac{|1-5|+|4-5|+|5-5|+|7-5|+|8-5|}{5}$
MAD = $\frac{4+1+0+2+3}{5}=\frac{10}{5}=2$
Variance (usually represented by ${\sigma}^2$) is an expression of how far the values in a data set are spread out from their mean. Variance is given by the formula:
${\sigma}^2=\frac{\sum_{i=1}^{n}(xi-\bar{x})^2}{n}$, where $\bar{x}$ is the arithmetic mean of the set of data, and $n$ is the number of elements in the set.
Standard Deviation (usually represented by $\sigma$) is the square root of the Variance. It is, therefore, given by the formula:
$\sigma=\sqrt{\frac{\sum_{i=1}^{n}(xi-\bar{x})^2}{n}}$, where $\bar{x}$ is the arithmetic mean of the data set, and $n$ is the number of elements in the set.
Suppose, for example, that we have data about the concentration of a chemical element in a large sample of a medication. The arithmetic mean (or just "mean") of this data set is 10 mg/ml and its standard deviation is 2 mg/ml. The following graph displays the distribution of concentration of the chemical element:
The Standard Deviation ($\sigma$) has the following properties:
68,2% of measurements are between $\bar{x}-\sigma$ and $\bar{x}+\sigma$. In our example, between $10-2$ and $10+2$ (between 8 and 12).
95,4% of measurements are between $\bar{x}-2\sigma$ and $\bar{x}+2\sigma$. In our example, between $10-2*2$ and $10+2*2$ (between 6 and 14).
99,7% of measurements are between $\bar{x}-3\sigma$ and $\bar{x}+3\sigma$. In our example, between $10-3*2$ and $10+3*2$ (between 4 and 16).
Important Definitions
Statistics is the discipline that studies the collection, modeling, analysis and interpretation of data.
Population in statistics is the set of ALL elements of interest. Typically, the population is very large. In an election, for example, the statistical population in a study could be the total number of votes (dozens of millions).
Sample in statistics is a subset collected from the statistical population.
Margin of Error is an expression of random sampling error. The larger the margin of error, the less confidence one should have on the conclusions drawn from the sample.
The margin of error (or standard error) of a percentage or proportion "p" is given by the formula:
$Margin=\frac{p(1-p)}{\sqrt{n}}$, where "n" is the sample size.
Frequency Distribution is a list, table or graph that displays all distinct outcomes in a sample and the number of times they occur.
For example, if we flip a coin 100 times, the frequency distribution could be 48 heads and 52 tails.
Probability of an Event is denoted by the ratio $P=r/n$, where "r" is the number of wanted outcomes, and "n" is the number of all possible outcomes.
For example, what is the probability that the outcome, when we flip a coin, is Heads? In this experiment, the number of wanted outcomes is 1 (Heads), and the total number of all possible outcomes is 2 (Heads or Tails). Therefore the probability that the outcome will be Heads is 1/2=50%
Solved SAT Practice Tests
Find Practice Tests in the following links:
SAT Practice Tests - Means, Medians, and Statistics
Additional Practice Tests - Means, Medians, and Statistics
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