MATH 1040 Chapter 3

An alternative method for finding a rough approximation for the quartiles

1 An alternative method for finding a rough approximation for the quartiles 2 never include the median in either half of the data. 3 find the median of the lower half of the data, which is the first quartile. 4The third quartile is the median of the upper half of the data.

Theorem: Empirical Rule for Bell-Shaped Distributions1

Approximately 68% of the data values lie within one standard deviation of the mean.

Theorem: Empirical Rule for Bell-Shaped Distributions2

Approximately 95% of the data values lie within two standard deviations of the mean.

Theorem: Empirical Rule for Bell-Shaped Distributions3

Approximately 99.7% of the data values lie within three standard deviations of the mean.

range

The easiest measure of dispersion to calculate is the range. The range of a data set is the difference between the largest and smallest values in the data set.

median

The median is the middle value in an ordered array of data.

Weighted Mean

The weighted mean is the mean of a data set in which each data value in the set does not hold the same relative importance

mean的Rounding Rule

When calculating the mean, round to one more decimal place than the largest number of decimal places given in the data. Occasional exceptions to this rule can be made when the type of data lends itself to a more natural rounding scheme, such as rounding values of currency to two decimal places.比最大小数位数多一位，除了 rounding values of currency to two decimal places

计算标准deviation的舍入规则Rounding Rule

When calculating the standard deviation, round to one more decimal place than the largest number of decimal places given in the data. Occasional exceptions to this rule can be made when the type of data lends itself to a more natural rounding scheme, such as rounding values of currency to two decimal places.

If we want to represent a five-number summary graphically, we can use a graph called

a box plot, "box-and-whisker plot

In a unimodal data set, if the mean, median, and mode are all equal, the shape of the distribution will be

bell-shaped.

larger data sets will result in a

better approximation

If exactly two values occur equally often, then the data set is said to be .

bimodal

a smaller standard deviation indicates that the data values lie

closer together.

data divided into 10 parts are called

deciles

the range is not as ? as other measures of dispersion.

descriptive

Since variance is a squared unit, we cannot use it to compare

distances between data values.

Note：The units on the x-axis in the graphs above are

distances from the mean in terms of numbers of standard deviations

IQR is small, and its median is one of the smallest as well, indicating that

fewer green candies are likely to appear in a bag.

Quartiles are used in a numerical description, aptly called the

five-number summary

coefficient of variation

for a set of data is the ratio of the standard deviation to the mean as a percentage. allows us to compare the spreads of data from two different sources, as shown in the following example.

of the three measures of center, the ？ is closest to the outlier while the median and mode are more similar in value and are not affected by the outlier.

mean

the standard deviation could never be a negative number, since

measures of distances are always positive.

if a quantitative data set has an outlier or is skewed, you should use the ？.

median

If more than two values occur equally often, the data set is

multimodal.

If all of the data values occur only once, or they each occur an equal number of times, we say that there is .

no mode

The median would not make sense for ？ despite the fact that it technically can be calculated for ordinal data.

nominal or ordinal data

data divided into 8 parts are called

octiles.

The "whiskers" are the lines that extend to

reach the minimum and maximum values.

sample mean 计算方法 x⎯⎯ : sample mean; read as " x-bar"

the arithmetic mean of a set of sample data

The interquartile range IQR is the difference between ?, It is the range of the middle 50% of the data.

the third quartile and the first quartile, and is seen as the size of the box in a box-and-whisker plot.

The mode is

the value in the data set that occurs most frequently.

when dividing a data set into four parts, we use ? quartiles.

three

If only one value occurs most often, then the data set is said to be .

unimodal

Note that if there is a mode, it will always be a

value in the data set.

a good rule of thumb,

一个很好的经验法则

algebra

代数

dispersion

分散

denominator

分母

booster

加速器

Quartiles

四分位数 If we divide a data set into four parts, the numbers that form the divisions are called quartiles.

Theorem

定理

mean

平均数

square

平方

square root

开平方根

percentile

百分位数又称百分位分数(percentile)，是一种相对地位量数，它是次数分布中的一个点。把一个次数分布排序后，分为100个单位，百分位数就是次数分布中相对于某个特定百分点的原始分数，它表明在次数分布中特定个案百分比低于该分数。

phosphorous

磷

coefficient

系数 相关关系是一种非确定性的关系，相关系数是研究变量之间线性相关程度的量。简单相关系数：又叫相关系数或线性相关系数，一般用字母P 表示，是用来度量变量间的线性关系的量。复相关系数：又叫多重相关系数。复相关是指因变量与多个自变量之间的相关关系。例如，某种商品的季节性需求量与其价格水平、职工收入水平等现象之间呈现复相关关系。典型相关系数：是先对原来各组变量进行主成分分析，得到新的线性关系的综合指标，再通过综合指标之间的线性相关系数来研究原各组变量间相关关系

Empirical Rule

经验法则

rounding scheme

舍入计划

diastolic blood pressure readings

舒张压读数

highwa miles per gallon (mpg)

高速公路每英里耗油量

subbasin

subbasin

给定135个值，求第10th percentile的value

1. 计算l ，2小数则进位到最大整数 3 在ordinal data里找整数到对应值

给定135个值，求第20 percentile的value

1. 计算l，2整数则进位到另一个邻近大整数，3 在 ordinal data里找到这两个整数对应的值 4 计算平均数

hinge.

If you do not include the median when you are approximating the first and third quartiles, you are actually calculating a completely different value, created by statistician John Tukey, called a hinge. The lower hinge is a rough approximation of the first quartile, and the upper hinge is a rough approximation of the third quartile.

bell-shaped

implies that there is symmetry about the middle value,

Calculating percentile，each of the methods could result

in a different answer, depending on the size and variation of your data set.

For this calculation to make sense, the data must be

interval or ratio

A standardized test score is often reported in terms of a

percentile, which is one measure of relative position.

mode is the best measure of center for

qualitative data.

mean 的 value only makes sense when the data is？

quantitative

We can compare two data values from different populations by comparing their.

respective percentiles，standard score, or z-score.

If the standard deviation equals 0, then none of the values differ from the mean, and therefore must all be the

same as the mean.

The greater the standard deviation, the more the data values are

spread out.

population mean, μ; Greek letter, mu

the mean of all the values in a population

A standard score tells us how far a value is from

the mean, specifically, how many standard deviations it is away from the mean.

IQR 更大 更高

the middle 50% of the data is more spread out and has larger values than those

what we might mean by the "average" college major

the most typical major=frequently occurring=mode