1 / 32

DESCRIPTIVE STATISTICS

DESCRIPTIVE STATISTICS. DESCRIPTIVE STATISTICS. Descriptive statistics are summary measures which define some important characteristics of data. Measures of location Measures of central tendency Measures of position Measures of dispersion (variation). Measure s of Location.

hanh
Download Presentation

DESCRIPTIVE STATISTICS

An Image/Link below is provided (as is) to download presentation Download Policy: Content on the Website is provided to you AS IS for your information and personal use and may not be sold / licensed / shared on other websites without getting consent from its author. Content is provided to you AS IS for your information and personal use only. Download presentation by click this link. While downloading, if for some reason you are not able to download a presentation, the publisher may have deleted the file from their server. During download, if you can't get a presentation, the file might be deleted by the publisher.

E N D

Presentation Transcript

  1. DESCRIPTIVE STATISTICS

  2. DESCRIPTIVE STATISTICS Descriptive statistics are summary measures which define some important characteristics of data. • Measures of location • Measures of central tendency • Measures of position • Measures of dispersion (variation)

  3. Measures of Location • Measures of central tendency Measures of central tendency are numerical values that tend to locate in some sense the middle of a set of data. • Geometric Mean • Harmonic Mean • Proportion • Arithmetic Mean  • Median  • Mode

  4. Sum of all the observations ( )divided by the number of the observations (n). Measures of central tendency Arithmetic Mean (Mean) The arithmetic mean is the most common measure of the central tendencyand is commonly used for symetrical distributions. It is used to summarize quantitative data.

  5. Example years Age distribution of seven children attending to a children clinic is given below {1,3,6,7,2,3,5}

  6. observation. Median value is Median The median is the middle value of the set of data when the data are ranked in order according to magnitude. When the data are put in order 50 % of the observations are less than or equal to the median, the rest is greater than the median .

  7. Example n is odd: 5, 28, 8, 10, 9 Ordered data 5, 8, 9, 10, 28 i =(5+1)/2=3 Median is 3rd value which is 9. n is even: 19, 20, 17, 27, 6, 21 Ordered data 6, 17, 19, 20, 21, 27 i=(6+1)/2=3.5 Median is halfway between the 3rd and 4th values, which is 19.5.

  8. Mode The mode is the value of x that occurs most frequently. • Data {1,3,7,3,2,3,6,7} • Mode : 3 • Data {1,3,7,3,2,3,6,7,1,1} • Mode : 1 and 3 • Data {1,3,7,0,2,-3, 6,5,-1} • Mode : No mode

  9. Suppose the age in years of the first 10 subjects enrolled in your study are:34, 24, 56, 52, 21, 44, 64, 44, 42, 46 Then the mean age of this group is 42.7 years To find the median, first order the data: 21, 24, 34, 42, 44, 44, 46, 52, 56, 64 The median is  (44+44)/2 = 44 years The mode is 44 years. Example

  10. Suppose the next patient enrolls and her age is 97 years. How does the mean, median and mode change? Ordered data: 21, 24, 34, 42, 44, 44, 46, 52, 56, 64, 97 Mean is 47,6 42,7 Median is 44 44 Mode is 44 44

  11. Comparison of Mean and Median • Mean is sensitive to “outliers” (a few very large or small values), so sometimes mean does not reflect the quantity desired. • 20, 21, 22, 23, 24, 25, 26, 90 87.5% of observations • Median is “resistant” to outliers • Median = 23.5 • Mean is attractive mathematically. • 50% of sample is above the median, 50% of • sample is below the median.

  12. or Geometric mean is a summary statistic useful when the measurement scale is not linear, it is computed as For example, in the area of psychometrics it is well known that the rated intensity of a stimulus (e.g., brightness of a light) is often a logarithmic function of the actual intensity of the stimulus (brightness measured in units of Lux). In this instance, the geometric mean is a better "summary" of ratings than the simple mean.

  13. For example, suppose you have an investment which earns 10% the first year, 50% the second year, and 30% the third year. What is its average rate of return? It is not the arithmetic mean, because what these numbers signify is that on the first year your investment was multiplied (not added to) by 1.10, on the second year it was multiplied by 1.50, and the third year it was multiplied by 1.30. The relevant quantity is the geometric mean of these three numbers, which is about 1.28966 or about 29% annual interest.

  14. For example, if a stock rose 10% in the first year, 20% in the second year and fell 15% in the third year, then we compute the geometric mean of the factors 1.10, 1.20 and 0.85 as (1.10 × 1.20 × 0.85)1/3 = 1.0391... and we conclude that the stock rose 3.91 percent per year, on average.

  15. Harmonic Mean is a "summary" statistic used in analyses of frequency data. The harmonic mean is sometimes used to average values that change in time. If a variable contains a zero (0) as a valid score, then the harmonic mean cannot be calculated (since it implies division by zero).

  16. In certain situations, the harmonic mean provides the correct notion of "average". For instance, if for half the distance of a trip you travel at 40 miles per hour and for the other half of the distance you travel at 60 miles per hour, then your average speed for the trip is given by the harmonic mean of 40 and 60, which is 48; that is, the total amount of time for the trip is the same as if you traveled the entire trip at 48 miles per hour. (Note however that if you had traveled for half the timeat one speed and the other half at another, the arithmetic mean, 50 miles per hour, would provide the correct notion of "average".)

  17. Proportion is a fraction in which the numerator is included within the denominator. A proportion is often expressed as a percentage. For example, we might describe the overall amount of smokers in a population as the proportion of people in the population who smoke. Suppose out of 125 people 15 smoke cigarette, then the proportion of people who smoke is P =

  18. Measures of position Measures of position are used to decribe the location of a specific piece of data in relation to the rest of the sample. Quartiles and percentiles are two most popular measures of position.

  19. Quartiles The first quartile, Q1, is a number such that at most one-fourth of the data are smaller in value than Q1 and at most three-fourths are larger. The second quartile, Q2, is the median. Quartiles are numeric values of variable that divide the ordered data into quarters; each set of data has three quartiles.

  20. The third quartile, Q3, is a number such that at most three-fourths of the data are smaller in value than Q3 and at most one-fourth are larger.

  21. Example: Birthweights of 24 infants are as follows:

  22. First quartile=Q1= th obs. is the fist quartile. 6th obs.=3100gr 7th obs.=3150gr 50x0.25=12.5gr Q1=3112.5gr. Second quartile=Median=Q2= th obs. is the second quartile or median. 12th obs.=3250gr 13th obs.=3250gr Q2=3250gr. Third quartile=Q3= th obs. is the third quartile

  23. 18th obs.=3600gr 19th obs.=3700gr 100x0.75=75gr. Q3=3675gr. 25% of the infants have birthweights less than 3112,5gr. Half of the infants have birthweights less than 3250gr. 75%of the infants have birthweights less than 3675 gr.

  24. Percentiles are numerical values of the variable that divide a set of ordered data into 100 equal parts; each set of data has 99 percentiles. The procedure for determining the value of any kth percentile involve three basic steps. • The data must be ordered • The position number i for the percentile in question must be determined. It is found by first calculating the value of .

  25. Example: Birth weights of 24 infants. • Find the 30th percentile. • When the data are ordered th observation, which is the half way between 7th (3150gr) and 8th (3200gr) observations, is the 30th percentile. The 30th percentile is therefore 3175gr, which means, 30% of the observations lie below 3175gr. • ii. Find the 50th percentile (median). • th observation, which is the half way between 12th (3250gr) and 13th observations (3250gr) is the 50th percentile. 3250gr is the 50th percentile. Half of the infants have birth weights less than 3250gr.

  26. Measures of dispersion Range The range is the simplest measure of dispersion. It is the difference between the highest valued (H) and the lowest valued (L) of the observations. Range= H-L

  27. Measures of dispersion Standard deviationis the average distance of observations to arithmetic mean. or Varianceis square of standard deviation. or

  28. Step 1 NOTE: The sum of the deviation, , is always zero. Step 3 Step 4 Step 2 Step 5

  29. The last set of data is more dispersed than the previous set, and therefore its variance is larger. s2=4.5 First sample Second sample s2=11.5

  30. Coefficient of variation (CV)shows the deviation as a percentage of arithmetic mean.

  31. s cv Example: Heights of ten subjects are measured in cm and m. Data are listed below Although standard deviations are different, coefficient of variations are the same.

  32. Interquartile Rangeis a measure of dispersion for non-symmetric data IQR = Q3 – Q1 Semi-Interquartile Range is used instead of standart deviation when distribution of data is non-symmetric. SIQR= (Q3 – Q1)/2 Ranked data, increasing order L Q1 Q2 Q3 H

More Related