Measures of Center 1 Measure of Center the
- Slides: 73
Measures of Center 1
Measure of Center the value at the center or middle of a data set 1. Mean 2. Median 3. Mode 4. Midrange (rarely used) 2
Mean Arithmetic Mean (Mean) the measure of center obtained by adding the values and dividing the total by the number of values What most of us call an average. 3
Notation ∑ denotes the sum of a set of values. x is the variable used to represent the individual data values. n represents the number of data values in a sample. N represents the number of data values in a population. 4
x is pronounced ‘x-bar’ and denotes the mean of a set of sample values ∑x x = n This is the sample mean µ is pronounced ‘mu’ and denotes the mean of all values in a population µ = ∑x N This is the population mean 5
Mean Advantages Is relatively reliable. Takes every data value into account Disadvantage Is sensitive to every data value, one extreme value can affect it dramatically; is not a resistant measure of center 6
Mean Example Major in Geography at University of North Carolina 7
Median the middle value when the original data values are arranged in order of increasing (or decreasing) magnitude often denoted by x~ (pronounced ‘x-tilde’) is not affected by an extreme value - is a resistant measure of the center 8
Finding the Median First sort the values (arrange them in order), then follow one of these rules: 1. If the number of data values is odd, the median is the value located in the exact middle of the list. 2. If the number of data values is even, the median is found by computing the mean of the two middle numbers. 9
Example 1 5. 40 1. 10 0. 42 0. 73 0. 48 1. 10 0. 66 10
Example 1 5. 40 1. 10 0. 42 0. 73 0. 48 1. 10 0. 66 Order from smallest to largest: 0. 42 0. 48 0. 66 0. 73 1. 10 5. 40 11
Example 1 5. 40 1. 10 0. 42 0. 73 0. 48 1. 10 0. 66 Order from smallest to largest: 0. 42 0. 48 exact middle 0. 66 0. 73 1. 10 5. 40 MEDIAN is 0. 73 12
Example 2 5. 40 1. 10 0. 42 0. 73 0. 48 1. 10 13
Example 2 5. 40 1. 10 0. 42 0. 73 0. 48 1. 10 Order from smallest to largest: 0. 42 0. 48 0. 73 1. 10 5. 40 14
Example 2 5. 40 1. 10 0. 42 0. 73 0. 48 1. 10 Order from smallest to largest: 0. 42 0. 48 0. 73 1. 10 5. 40 Middle values 15
Example 2 5. 40 1. 10 0. 42 0. 73 0. 48 1. 10 Order from smallest to largest: 0. 42 0. 48 0. 73 1. 10 5. 40 Middle values 0. 73 + 1. 10 2 = 0. 915 16
Example 2 5. 40 1. 10 0. 42 0. 73 0. 48 1. 10 Order from smallest to largest: 0. 42 0. 48 0. 73 1. 10 5. 40 Middle values 0. 73 + 1. 10 2 = 0. 915 MEDIAN is 0. 915 17
Mode the value that occurs with the greatest frequency Data set can have one, more than one, or no mode Bimodal two data values occur with the same greatest frequency Multimodal more than two data values occur with the same greatest frequency No Mode no data value is repeated 18
Mode - Examples a. 5. 40 1. 10 0. 42 0. 73 0. 48 1. 10 b. 27 27 27 55 55 55 88 88 99 c. 1 2 3 6 7 8 9 10 19
Mode - Examples a. 5. 40 1. 10 0. 42 0. 73 0. 48 1. 10 Mode is 1. 10 b. 27 27 27 55 55 55 88 88 99 Bimodal - c. 1 2 3 6 7 8 9 10 No Mode 27 & 55 20
Definition Midrange the value midway between the maximum and minimum values in the original data set Midrange = maximum value + minimum value 2 21
Midrange Sensitive to extremes because it uses only the maximum and minimum values. Midrange is rarely used in practice 22
Round-off Rule for Measures of Center Carry one more decimal place than is present in the original set of values. 23
Common Distributions 24
Skewed and Symmetric distribution of data is symmetric if the left half of its histogram is roughly a mirror image of its right half Skewed distribution of data is skewed if it is not symmetric and extends more to one side than the other 25
Symmetry and skewness 26
Measures of Variation 27
Measures of Variation spread, variability of data width of a distribution 1. Standard deviation 2. Variance 3. Range (rarely used) 28
Standard deviation The standard deviation of a set of sample values, denoted by s, is a measure of variation of values about the mean. 29
Sample Standard Deviation Formula Σ (x – x) n– 1 2 s= 30
Sample Standard Deviation (Shortcut Formula) s= nΣ ( x ) – (Σx) n (n – 1) 2 2 31
Population Standard Deviation σ = Σ (x – µ) 2 N σ is pronounced ‘sigma’ This formula only has a theoretical significance, it cannot be used in practice. 32
Example Values: 1, 3, 14 • Find the sample standard deviation: • Find the population standard deviation: 33
Example Values: 1, 3, 14 • Find the sample standard deviation: • s = 7. 0 • Find the population standard deviation: • σ = 5. 7 34
Variance The variance is a measure of variation equal to the square of the standard deviation. Sample variance: s 2 - Square of the sample standard deviation s Population variance: σ2 - Square of the population standard deviation σ 35
Variance - Notation s = sample standard deviation s 2 = sample variance σ = population standard deviation σ 2 = population variance 36
Example Values: 1, 3, 14 s = 7. 0 s 2 = 49. 0 σ = 5. 7 σ2 = 32. 7 37
Range (Rarely used) The difference between the maximum data value and the minimum data value. Range = (maximum value) – (minimum value) It is very sensitive to extreme values; therefore range is not as useful as the other measures of variation. 38
Using Excel 39
Using Excel Enter values into first column 40
Using Excel In C 1, type “=average(a 1: a 6)” 41
Using Excel Then, Enter 42
Using Excel Same thing with “=stdev(a 1: a 6)” 43
Using Excel Same with “=median(a 1: a 6)” - and add some labels 44
Using Excel Same with min, max, and mode 45
Usual and Unusual Events 46
Usual values in a data set are those that are typical and not too extreme. Maximum usual value = (mean) + 2 * (standard deviation) Minimum usual value = (mean) – 2 * (standard deviation) 47
Usual values in a data set are those that are typical and not too extreme. 48
Rule of Thumb Based on the principle that for many data sets, the vast majority (such as 95%) of sample values lie within two standard deviations of the mean. A value is unusual if it differs from the mean by more than two standard deviations. 49
Empirical (or 68 -95 -99. 7) Rule For data sets having a distribution that is approximately bell shaped, the following properties apply: About 68% of all values fall within 1 standard deviation of the mean. About 95% of all values fall within 2 standard deviations of the mean. About 99. 7% of all values fall within 3 standard deviations of the mean. 50
The Empirical Rule 51
The Empirical Rule 52
The Empirical Rule 53
Measures of Relative Standing 54
Z-score (or standardized value) The number of standard deviations that a given value x is above or below the mean 55
Measure of Position: Z-score Sample x – x z= s Population x – µ z= σ Round z scores to 2 decimal places 56
Interpreting Z-scores Whenever a value is less than the mean, its corresponding z score is negative Ordinary values: – 2 ≤ Z-score ≤ 2 Unusual values: Z-score < – 2 or Z-score > 2 57
Percentiles Measures of location. There are 99 percentiles denoted P 1, P 2, . . . P 99, which divide a set of data into 100 groups with about 1% of the values in each group. 58
Finding the Percentile of a Data Value Percentile of value x = number of values less than x total number of values • 100 Round it off to the nearest whole number 59
Example 2, pg 116 35 sorted values: 4. 5 5 6. 5 7 20 20 29 30 35 40 40 41 50 52 52 60 65 68 68 70 70 72 74 75 80 100 113 116 120 125 132 150 160 200 225 Find the percentile of 29 60
Example 2, pg 116 35 sorted values: 4. 5 5 6. 5 7 20 20 29 30 35 40 40 41 50 52 52 60 65 68 68 70 70 72 74 75 80 100 113 116 120 125 132 150 160 200 225 Find the percentile of 29 Percentile of 29 = 17 (rounded) 61
Converting from the kth Percentile to the Corresponding Data Value Notation L= k 100 • n n total number of values in the data set k percentile being used L locator that gives the position of a value Pk kth percentile 62
Example 3, pg 116 35 sorted values: 4. 5 5 6. 5 7 20 20 29 30 35 40 40 41 50 52 52 60 65 68 68 70 70 72 74 75 80 100 113 116 120 125 132 150 160 200 225 Find P 60 63
Example 3, pg 116 35 sorted values: 4. 5 5 6. 5 7 20 20 29 30 35 40 40 41 50 52 52 60 65 68 68 70 70 72 74 75 80 100 113 116 120 125 132 150 160 200 225 Find P 60 = 71 64
Converting from the kth Percentile to the Corresponding Data Value 65
Quartiles Measures of location, denoted Q 1, Q 2, and Q 3, which divide a set of data into four groups with about 25% of the values in each group. Q 1 (First Quartile) separates the bottom 25% of sorted values from the top 75%. Q 2 (Second Quartile) same as the median; separates the bottom 50% of sorted values from the top 50%. Q 3 (Third Quartile) separates the bottom 75% of sorted values from the top 25%. 66
Quartiles To calculate the quartile for homework and other Course. Compass work, using Excel: 1. Sort the data 2. Enter =quartile(<range>, 1) 3. Find the result in the sorted data 4. If the result is not in the sorted data, go to the next higher value 67
Example - Quartile 4. 5 5 6. 5 7 20 20 29 30 35 40 40 41 50 52 52 60 65 68 68 70 70 72 74 75 80 100 113 116 120 125 132 150 160 200 225 =quartile(A 1: G 5, 1) give 37. 5 is between 35 and 40 The 1 st quartile value is 40 68
Quartiles Q 1 , Q 2, Q 3 divide ranked scores into four equal parts 25% 25% (minimum) Q 1 Q 2 Q 3 (maximum) (median) 69
Some Other Statistics Interquartile Range (or IQR): Q 3 – Q 1 Semi-interquartile Range: Midquartile: Q 3 + Q 1 Q 3 – Q 1 2 2 10 - 90 Percentile Range: P 90 – P 10 70
5 -Number Summary For a set of data, the 5 -number summary consists of the ● minimum value ●first quartile Q 1 ●median (or second quartile Q 2) ●third quartile, Q 3 ●maximum value. 71
Example 35 sorted values: 4. 5 5 6. 5 7 20 20 29 30 35 40 40 41 50 52 52 60 65 68 68 70 70 72 74 75 80 100 113 116 120 125 132 150 160 200 225 Find the 5 -number summary 72
Example Min = 4. 5 Q 1 = 40 Median = 50 Q 3 = 1130 Max = 225 73
- Anova repeated measures
- Measures of location statistics
- Gibbons jacobean city comedy download
- Air temperature
- Measures of center and spread worksheet
- Hát kết hợp bộ gõ cơ thể
- Lp html
- Bổ thể
- Tỉ lệ cơ thể trẻ em
- Chó sói
- Thang điểm glasgow
- Alleluia hat len nguoi oi
- Các môn thể thao bắt đầu bằng tiếng đua
- Thế nào là hệ số cao nhất
- Các châu lục và đại dương trên thế giới
- Cong thức tính động năng
- Trời xanh đây là của chúng ta thể thơ
- Mật thư anh em như thể tay chân
- 101012 bằng
- Phản ứng thế ankan
- Các châu lục và đại dương trên thế giới
- Thể thơ truyền thống
- Quá trình desamine hóa có thể tạo ra
- Một số thể thơ truyền thống
- Cái miệng bé xinh thế chỉ nói điều hay thôi
- Vẽ hình chiếu vuông góc của vật thể sau
- Nguyên nhân của sự mỏi cơ sinh 8
- đặc điểm cơ thể của người tối cổ
- Thế nào là giọng cùng tên?
- Vẽ hình chiếu đứng bằng cạnh của vật thể
- Fecboak
- Thẻ vin
- đại từ thay thế
- điện thế nghỉ
- Tư thế ngồi viết
- Diễn thế sinh thái là
- Dot
- Số nguyên tố là gì
- Tư thế ngồi viết
- Lời thề hippocrates
- Thiếu nhi thế giới liên hoan
- ưu thế lai là gì
- Hươu thường đẻ mỗi lứa mấy con
- Sự nuôi và dạy con của hổ
- Sơ đồ cơ thể người
- Từ ngữ thể hiện lòng nhân hậu
- Thế nào là mạng điện lắp đặt kiểu nổi
- Semi interquartile range
- Measure of center
- Measure of center
- If mean greater than median
- It measure the vertical distance
- Using statistical measures to compare populations
- Measures of concentration molarity quiz
- Costs of unemployment
- What are cms core measures
- Rumus kurtosis
- Ukuran asosiasi dalam epidemiologi
- Weights and measure training
- Technical performance parameters
- Balanced scorecard measures that drive performance
- Technical performance measures systems engineering
- 4 d's of crime prevention
- Strategic measures
- Numerical measures
- Homework 7 segment lengths
- Two tangents intersecting outside the circle
- Secants and tangents
- Andy field repeated measures anova
- Multivariate anova spss
- Repeated measures design
- Repeated-measures design
- Repeated measures manova
- Raleigh and rosse real company