Statistics is a discipline that studies the collection, analysis and interpretation of data, and it plays an important role in data analysis. As a powerful programming language, Python has a wide range of applications in the field of data analysis. This article will introduce important statistical concepts in Python data analysis to help you better understand and apply statistical knowledge.
1. Data type
1.1 Numeric data
Numeric data refers to data types that represent values or sizes, including integers, floating-point numbers, and complex numbers. In Python, you can use the NumPy library to process numerical data, such as performing numerical calculations and statistical analysis.
1.2 Category data
Categorical data refers to data types representing categories or labels, including nominal variables and ordinal variables, etc. In Python, the pandas library can be used to process categorical data, such as data cleaning and feature encoding.
1.3 Temporal data
Temporal data refers to data types that represent time or date, such as year, month, and specific time points. In Python, you can use the datetime library to process time-type data, such as time series analysis and date calculation.
2. Descriptive statistics
Descriptive statistics are statistical methods for summarizing and describing data sets. Python provides a wealth of descriptive statistics tools and functions that can help us calculate the central tendency, degree of dispersion, and distribution characteristics of data.
2.1 Central tendency
The central tendency refers to the measurement of the central position of the data set. Commonly used indicators include mean, median, and mode. Using functions in the pandas and NumPy libraries, we can easily compute these metrics.
- 均值（mean）：所有数据的平均值。使用`DataFrame.mean()`或`np.mean()`函数计算。
- 中位数（median）：将数据按照大小排序后，位于中间的数值。使用`DataFrame.median()`或`np.median()`函数计算。
- 众数（mode）：数据集中出现次数最多的数值。使用`DataFrame.mode()`或`scipy.stats.mode()`函数计算。
2.2 Discrete degree
The degree of dispersion refers to the measurement of the degree of dispersion of the data set. Commonly used indicators include standard deviation, variance, and interquartile range. Using functions in the pandas and NumPy libraries, we can easily calculate these metrics.
- 标准差（standard deviation）：数据集各个数据与均值之差的平方和的平均值的平方根。使用`DataFrame.std()`或`np.std()`函数计算。
- 方差（variance）：数据集各个数据与均值之差的平方和的平均值。使用`DataFrame.var()`或`np.var()`函数计算。
- 四分位数范围（interquartile range）：数据集上下四分位数之差，表示数据中间50%的变动范围。使用`DataFrame.quantile()`函数计算。
2.3 Distribution characteristics
Distribution characteristics refer to the description of the distribution form of a data set. Commonly used indicators include skewness, kurtosis, and frequency statistics. Using functions in the pandas, SciPy, and matplotlib libraries, we can easily calculate and visualize these metrics.
- 偏度（skewness）：数据分布的偏斜程度。使用`DataFrame.skew()`或`scipy.stats.skew()`函数计算。
- 峰度（kurtosis）：数据分布的尖锐程度。使用`DataFrame.kurtosis()`或`scipy.stats.kurtosis()`函数计算。
- 频数统计（frequency count）：数据集中各个唯一数值的出现次数统计。使用`DataFrame.value_counts()`函数计算。
3. Probability distribution
A probability distribution is a function that describes the probability of a random variable taking a value. Commonly used probability distributions include normal distribution, binomial distribution, and Poisson distribution. In Python, the SciPy library can be used for modeling and analysis of probability distributions.
3.1 Normal distribution
The normal distribution (also known as the Gaussian distribution) is one of the most common probability distributions, and it exhibits a bell-shaped curve. Using functions in the SciPy library, we can generate normally distributed random numbers, calculate probability densities and cumulative distributions, and more.
- 生成随机数：使用`scipy.stats.norm.rvs()`函数生成服从正态分布的随机数。
- 计算概率密度：使用`scipy.stats.norm.pdf()`函数计算指定取值点的概率密度。
- 计算累积分布：使用`scipy.stats.norm.cdf()`函数计算指定取值点的累积分布。
3.2 Binomial distribution
The binomial distribution is a probability distribution that describes repeated binary trials, such as the outcomes of coin tosses. Using functions in the SciPy library, we can calculate the probability mass of the binomial distribution, cumulative distribution, random sampling, etc.
- 计算概率质量：使用`scipy.stats.binom.pmf()`函数计算指定取值的概率质量。
- 计算累积分布：使用`scipy.stats.binom.cdf()`函数计算指定取值的累积分布。
- 生成随机数：使用`scipy.stats.binom.rvs()`函数生成符合二项分布的随机数。
3.3 Poisson distribution
The Poisson distribution is a probability distribution that describes the number of occurrences of an event per unit time, such as the number of calls received within a unit time. Using functions in the SciPy library, we can calculate the probability mass of the Poisson distribution, cumulative distribution, and random sampling, among others.
- 计算概率质量：使用`scipy.stats.poisson.pmf()`函数计算指定取值的概率质量。
- 计算累积分布：使用`scipy.stats.poisson.cdf()`函数计算指定取值的累积分布。
- 生成随机数：使用`scipy.stats.poisson.rvs()`函数生成符合泊松分布的随机数。
4. Hypothesis testing
Hypothesis testing is a method used to perform inferential statistical analyzes on data sets, such as comparing whether sample means are significantly different. In Python, the SciPy library can be used to perform hypothesis testing and help us draw statistically significant conclusions.
4.1 One-sample hypothesis testing
One-sample hypothesis testing is used to test whether there is a significant difference between the parameters of a single sample and known values. Common hypothesis testing includes one-sample t-test and one-sample Z-test. Using functions in the SciPy library, we can perform these hypothesis tests.
- 单样本t检验：使用`scipy.stats.ttest_1samp()`函数进行单样本t检验。
- 单样本Z检验：使用`scipy.stats.zscore()`函数计算样本标准差，然后与已知值进行比较。
4.2 Two-sample hypothesis testing
The two-sample hypothesis test is used to test whether there are significant differences in the parameters of two independent samples. Common hypothesis tests include independent sample t-test and Mann-Whitney U test. Using functions in the SciPy library, we can perform these hypothesis tests.
- 独立样本t检验：使用`scipy.stats.ttest_ind()`函数进行独立样本t检验。
- Mann-Whitney U检验：使用`scipy.stats.mannwhitneyu()`函数进行Mann-Whitney U检验。
4.3 Correlation test
The correlation test is used to test whether there is a significant linear correlation between two variables. Common hypothesis tests include Pearson correlation coefficient test and Spearman rank correlation coefficient test. Using functions in the SciPy library, we can perform these hypothesis tests.
- Pearson相关系数检验：使用`scipy.stats.pearsonr()`函数进行Pearson相关系数检验。
- Spearman秩相关系数检验：使用`scipy.stats.spearmanr()`函数进行Spearman秩相关系数检验。
in conclusion
Through this introduction, you learned important statistical concepts in Python data analysis, including data types, descriptive statistics, probability distributions, and hypothesis testing. These concepts provide you with fundamental theory and methods in the data analysis process. Of course, in addition to the content introduced in this article, statistics also includes more knowledge and techniques, which require your continuous study and practice.
In practical applications, please choose the appropriate statistical methods and tools according to your specific needs and data characteristics. At the same time, it is also important to rationally interpret and interpret statistical results.