3.2.4. Statistical description of measurement data

1.1.1.1. Description of the centralized trend (central tendency)

The main statistical indicators that describe the trend are arithmetic averages, geometric averages, median numbers, which are also known as positional metrics (measures of location) 1.1.1.1.1. Arithmetic averages (arithmetic mean)

The arithmetic mean is applicable to the frequency distribution symmetrical data. In the case of outliers, or when the frequency distribution is asymmetric, it is not appropriate to choose the average water level of the arithmetic mean to describe the data.

(1) Generally, the overall mean is expressed in μ, the sample is represented by a symbol, observing n individuals, X is the observation value, the formula for the mean number is:

(2) When the sample size is relatively large, if the mean is calculated by the frequency tables, the formula (the formula of the mean weighted calculation) is as follows:

In the formula: F is the frequency of each group of segments, 1.1.1.1.2 for the group median of the corresponding group segment. Geometric averages (geometric mean,g)

Geometric averages can only be applied to right-biased distribution data, not to left-biased distribution data

(1) Generally, the geometric mean is equal to the nth root of the product of all n observations of a variable. The formula is:

In the formula: to calculate the logarithm of x, its calculation can be based on the base of 10 (recorded as LG), can also be used as the natural number E as the bottom (denoted as LN)

(2) When the sample volume is large, if the mean is calculated by the frequency tables, the formula (the geometric mean weighted formula) is as follows:

1.1.1.1.3. Median (median,m)

Refers to all n observations of a variable in the order of size, in the middle of the value, recorded as M, the formula is as follows:

① median is not sensitive to outliers

② when the data is symmetrically distributed, the mean and median are close, and when the data is in the right-biased distribution, the mean is greater than the median, and when the data is left-biased, the mean is less than the median; 1.1.1.1.4. The majority of 1.1.1.2. Description of the discrete trend (dispersion)

The discrete trend refers to the degree to which all observations of the measurement data deviate from the central position, also known as the variance metric (measures of variation). The main statistical indicators describing discrete trends are the total, the interval, variance, standard deviation, and variation coefficients, also known as positional metrics (measures of location) 1.1.1.2.1. Total Pitch (Rang,r)

1.1.1.2.2. Number of percentiles (quartile) and percentile spacing, percentile (percentile)

The ① (quartile) is a numeric value between the maximum and minimum values, and the difference of two is called the spacing of the bits.

② percentile (percentile) is a positional indicator, expressed by. It means that the number of observations on the left side (i.e. less than the side) of the sequence in ascending order is the percentage of the entire sample. The formula is:

In the formula: for the percentile, L is the lower limit of the group, I is the group of the group spacing, is the frequency of the group, n is the total frequency, is the group of previous groups of the cumulative frequency of the segment.

③ statistics will be a special 3 sub-number, and collectively referred to as four (quartile), and respectively known as the first, 24 or three or four cent, recorded as, and, and the difference is four sub-spacing (quartile range,q), the formula is:

Example:

Group	Values in Group	Frequency	Cumulative frequency
0~5	2.5	1	1
5~10	7.5	2	3
10~15	12.5	4	7
15~20	17.5	6	13
20~25	22.5	7	20
25~30	27.5	9	29
30~35	32.5	13	42
35~40	37.5	23	65
40~45	42.5	34	99
45~50	47.5	2	101

1.1.1.2.3. Variance (VARIANCE,S2)

Variance is an indicator that describes the average degree of dispersion of all observations and averages, expressed in general.

(1) in general,

(2) When the sample size is larger,

1.1.1.2.4. Merge variance (polled variance) 1.1.1.2.5. Standard deviation (DEVIATION,SD)

Standard deviation is an indicator that describes the average degree of dispersion of all observations and averages of a variable, and typically uses s to denote standard deviation of the sample.

(1) in general,

(2) When the sample size is larger,

The standard deviation measure unit and the original variable unit of measure are consistent, for the same unit of measurement, the standard deviation is larger, the data is more discrete. 1.1.1.2.6. Merge standard deviation (polled standardized deviation) 1.1.1.2.7. Coefficient of variability (coefficient of VARIATION,CV)

The coefficient of variation is a measure of the relative degree of dispersion, which is calculated as:

Variation coefficient is dimensionless index, can be used to compare several dimensions of different variables of the difference between the degree of divergence, but also can be used to compare the same dimension but the difference between a few variables of the divergence between the differences.

3.2.4. Statistical description of measurement data