八.描述统计(descriptive statistics)

1.平均数(average or arithmetic mean)

2.中数(median)

To calculate the median of n numbers，first order the numbers from least to greatest；if n is odd，the median is defined as the middle number，while if n is even，the median is defined as the average of the two middle numbers. For the data 6, 4, 7, 10, 4, the numbers, in order, are 4, 4, 6, 7, 10, and the median is 6, the middle number. For the numbers 4, 6, 6, 8, 9, 12, the median is (6+8 )/2 = 7. Note that the mean of these numbers is 7.5.

3.众数(mode)：一组数中的众数是指出现频率最高的数。

例：the mode of 7，9，6，7，2，1 is 7。

4.值域(range)：表明数的分布的量，其被定义为最大值减最小值的差。

例：the range of–1，7，27，27，36 is 36-(-1)= 37。

5.标准方差(standard deviation)：

One of the most common measures of dispersion is the standard deviation. Generally speaking, the greater the data are spread away from the mean, the greater the standard deviation. The standard deviation of n numbers can be calculated as follows:

(1)find the arithmetic mean ;

(2)find the differences between the mean and each of the n numbers ;

(3)square each of the differences ;

(4)find the average of the squared differences ;

(5)take the nonnegative square root of this average.

Notice that the standard deviation depends on every data value, although it depends most on values that are farthest from the mean. This is why a distribution with data grouped closely around the mean will have a smaller standard deviation than data spread far from the mean.

6.排列与组合

There are some useful methods for counting objects and sets of objects without actually listing the elements to be counted. The following principle of Multiplication is fundamental to these methods.

If a first object may be chosen in m ways and a second object may be chosen in n ways, then there are mn ways of choosing both objects.

As an example, suppose the objects are items on a menu. If a meal consists of one entree and one dessert and there are 5 entrees and 3 desserts on the menu, then 5×3 = 15 different meals can be ordered from the menu. As another example, each time a coin is flipped, there are two possible outcomes, heads and tails. If an experiment consists of 8 consecutive coin flips, the experiment has 28 possible outcomes, where each of these outcomes is a list of heads and tails in some order.

☆阶乘：factorial notation

假如一个大于1的整数n，计算n的阶乘被表示为n!，被定义为从1至n所有整数的乘积，

例如：4! = 4×3×2×1= 24

注意：0! = 1! = 1