POPULATION SAMPLE GROUP INDIVIDUAL ORDINAL DATA NOMINAL DATA

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1 NUMERICAL DATA POPULATION ALPHANUMERIC DATA INDIVIDUAL ORDINAL DATA SAMPLE GROUP NOMINAL DATA

2 Lecture 2 - Statistical indicators Minimum, maximum Mean or Average Standard deviation Median Quartiles Percentiles, deciles

3 Data table No. Year Name Surname Sex Area Age Decade Occupation Stage CALOTA LUCIA F RURAL FARA OCUPATIE III CONSTANTIN MARIN M URBAN FARA OCUPATIE III FLOREA ELENA F RURAL PENSIONAR II HOLT MARIANA F URBAN PENSIONAR I IVANESCU VIRGIL M RURAL PENSIONAR II LEPADAT MARIN M URBAN PENSIONAR III MANOLACHE EUGENIA F RURAL SALARIAT IV MARINESCU DAN M RURAL PENSIONAR IV STAN SANDU M URBAN PENSIONAR V NEAGU MARIA M URBAN PENSIONAR III NEDELEA GHEORGHE F RURAL PENSIONAR II ORZESCU ION M URBAN PENSIONAR V PALIU MARIN F RURAL PENSIONAR IV PISICA MIHAIL F RURAL PENSIONAR III POPESCU PETRE M URBAN PENSIONAR IV PREDA ION M RURAL SALARIAT V ALBU NICOLAE M RURAL SALARIAT V RADUCAN ELISABETA M URBAN FARA OCUPATIE IV RADUCEANU ION M URBAN FARA OCUPATIE III IONESCU MARIA M URBAN FARA OCUPATIE IV

4 Statistical indicators for data series If we record the numerical values of a parameter for several individuals, we create a DATA SERIES Hb: 12,5; 13,5; 15,3; 16,4; 11,7,...etc Age: 36; 54; 73; 46; 31; 46;...etc We use the following mathematical notation: X: x 1, x 2, x 3,...x n Y: y 1, y 2, y 3,...y m

5 Minimum and maximum X: 58, 74, 70, 71, 56, 68, 70, 82, 62, 62 (ages) Minimum=56, Maximum=82 Y: 58, 74, 70, 71, 56, 68, 70, 82, 62, 62, 59, 46, 57, 71 (4 more values added to the first series) Minimum=46, Maximum=82 Range of values R = Max Min = 36

6 Arithmetic mean THE MEAN OR AVERAGE VALUE of a statistical data series is the sum of all the values divided by the number of values in that series. The mean is an indicator of the central tendency of the series, and usually shows where data tend to gather. Although it is computed using real, measured values, the average is an abstract number.

7 Arithmetic mean - the influence of outliners (extreme values) Y is made of the values of X plus 4 more values. The value 46, extremely small, greatly modifies the average.

8 Weighted arithmetic mean/ Weighted average F 1, F 2,...,F m are the frequencies of apparition in the series (weights) of the values x 1,x 2,...,x m. Other means - geometric mean, harmonic mean - are less widely used. Children No.of families No. of children Total Average no. of child per family

9 Variance The variance is a measurement of the spread of values in a data series Observations: 1. For series with almost equal averages, the data with the higher variance is more spread. 2. For series with almost equal variances, the data with the lower average is more spread.

10 Standard Deviation Variance has the following disadvantages: 1. It is measured with the squared units of the original variable meaningless units 2. Generally, it has very high values compared with the average. Therefore, another indicator, called the standard deviation, is commonly used, and it is the square root of variance. We note standard deviation as σ and it has the following formula:

11 Practical example Let us suppose we measured blood pressure in two patients, daily, for 10 days, obtaining the following values for systolic (maximum) blood pressure: 170, 180, 160, 180, 190, 190, 180, 190, 170, 190, for the first patient 160, 170, 190, 160, 190, 190, 200, 180, 180, 180, for the second patient

12 Computations Because both series have the same average value 180, we can conclude that Y series is more spread, having the higher standard deviation, 13.3, compared to X s standard deviation

13 Coefficient of variation It is the ratio between standard deviation and average, if the average differs from 0, andismeasuredasapercentage: For the previous series, the coefficient of variation is greater for the one that is more spread, that is for the one with the higher standard deviation: 1. C.V. x = 10,5 / 180 = 0,058 = 5,8 %. 2. C.V. y = 13,3 / 180 = 0,073 = 7,3%.

14 Asymmetry indicators Median is the middle number, in a sorted list of values. Half of the values are lower than the median and half are higher. For a measured parameter (e.g. weight, height, age), it represents the value located exactly in the middle of the data series, if individuals are ordered ascending or descending. Q1 quartile for a measured parameter, it represents the value to which a quarter of individuals have lower values and three quarters - higher values. Q3 quartile is the value to which three quarters of individuals have lower values and a quarter higher values.

15 Median example Systolic blood pressure 10 days measurements 150,160,160, 170,160,170,150,160,170,160 First we order the values in increasing order: 150,150,160,160,160,160,160,170,170,170 In this case, the median is between the fifth and sixth value in the ordered sequence, i.e. 160 If these two middle values are different, we consider the median as their arithmetic mean. If we have an odd number of measurements then the median is the value found in the middle of the ordered series.

16 Other statistical indicators Deciles. Used for greater samples, including hundreds of individuals. There are nine deciles, each corresponding to 10%, 20%... 90% of the sample, in a similar way to quartiles. The 5th decile is the median. Percentiles. Used in studies on thousands of cases, usually of larger interest, national or international. The percentiles correspond to values greater than 1%, 2%,..., 99% of the values found in the sample group. The 50th percentile is the median. Mode is the most frequently occurring number found in a series of values. E.g is the mode since it appears more times than any other number in the set: 3, 3, 6, 9, 16, 16, 16, 27, 27, 37, 48.