Stat Lect1

‫اﻻﺣﺼﺎء اﻻﺳﺘﺪﻻﻟﻰ‬ ‫‪ -1‬ﻣﺮاﺟﻌﺔ أهﻢ اﻟﻤﻔﺎهﻴﻢ‬

Population

Simple Random

Sample

X Q R H E A C S Z L N Y T M P V G U F J I W D B K O

Population

A F G P T

Stratified Random

Sample

B C R

D

E

25%

H

I K L 50% N O X U Z Y 25% Q V S W

25% 50% 25%

Population

AB OP

CD QR

MN

Sample

EFG HI

JKL

Cluster Random

STU

AB

Population

OP

CD QR

MN

STU EFG HI

JKL

B

I F

G

Two- Stage Random

Sample

Population

Systematic Non-Random

Sample

A F L Q V

B CDE G HIJ MKNO RPST WU X Y

Population Easily Accessible

Convenience Non-Random

Sample

X Q R H E A C S N Y T Z L M P V G U F J I W D B K O

Population Especially Qualified

Purposive Non-Random

Sample

X Q R A C Z L N V P G J B W O

H E S Y T M U F I D K

‫أﺧﻄﺎء اﻟﻌﻴﻨﺎت‬ ‫) اﻟﻔﺮق ﺑﻴﻦ إﺣﺼﺎءات اﻟﻌﻴﻨﺔ وﻣﻌﺎﻟﻢ اﻟﻤﺠﺘﻤﻊ (‬

‫أﺧﻄﺎء ﺗﺤﻴﺰ‬

‫ﺗﺤﻴﺰ ﻓﻰ اﺧﺘﻴﺎر‬ ‫اﻟﻌﻴﻨﺔ ﻣﺜﻞ ﺗﺒﺪﻳﻞ‬ ‫وﺣﺪات ﻣﻌﺎﻳﻨﺔ ﻣﺤﻞ‬ ‫أﺧﺮى ‪ ،‬واﻟﻔﺸﻞ ﻓﻰ‬ ‫اﻟﺤﺼﻮل ﻋﻠﻰ‬ ‫ﺑﻴﺎﻧﺎت ﺟﺰء آﺒﻴﺮ‬ ‫ﻣﻦ اﻟﻌﻴﻨﺔ‬

‫أﺧﻄﺎء ﻣﻌﺎﻳﻨﺔ‬ ‫ﺘﺤﻴﺯ ﻓﻰ ﺤﺴﺎﺏ‬ ‫ﺍﻟﺘﻘﺩﻴﺭﺍﺕ ﻨﺘﻴﺠﺔ‬ ‫ﻋﺩﻡ ﺍﺴﺘﺨﺩﺍﻡ‬ ‫ﺍﻷﺴﺎﻟﻴﺏ ﺍﻹﺤﺼﺎﺌﻴﺔ‬ ‫ﺍﻟﻤﻨﺎﺴﺒﺔ‬

‫ﺍﺴﺘﺨﺩﺍﻡ ﺒﻴﺎﻨﺎﺕ ﺍﻟﻌﻴﻨﺔ‬ ‫ﺒﺩﻻ ﻤﻥ ﺍﻟﻤﺠﺘﻤﻊ ﻭﺘﻌﺘﻤﺩ‬ ‫ﻫﺫﻩ ﺍﻷﺨﻁﺎﺀ ﻋﻠﻰ ﺤﺠﻡ‬ ‫ﺍﻟﻌﻴﻨﺔ ﻭﺘﺒﺎﻴﻥ ﺍﻟﻤﺠﺘﻤﻊ‬ ‫ﻭﻁﺭﻕ ﺍﺨﺘﻴﺎﺭ ﺍﻟﻌﻴﻨﺔ‬

‫ﻧﻔﺮض أن ﻟﺪﻳﻨﺎ ﻣﺠﺘﻤﻊ ﻳﺘﻜﻮن ﻣﻦ أرﺑﻊ درﺟﺎت )‬ ‫‪ (2,4,6,8‬وأردﻧﺎ اﺧﺘﻴﺎر ﻋﻴﻨﺎت ﺗﺸﻤﻞ آﻞ ﻣﻨﻬﺎ‬ ‫وﺣﺪﺗﻴﻦ أو درﺟﺘﻴﻦ ‪ ،‬ﻓﺈن آﻞ اﻟﻌﻴﻨﺎت اﻟﻌﺸﻮاﺋﻴﺔ‬ ‫اﻟﻤﻤﻜﻦ اﻟﺤﺼﻮل ﻋﻠﻴﻬﺎ ﻣﻦ ذﻟﻚ اﻟﻤﺠﺘﻤﻊ هﻰ‪:‬‬

‫„‬ ‫„‬ ‫„‬ ‫„‬

‫‪2،2‬‬ ‫‪2،4‬‬ ‫‪2،6‬‬ ‫‪2،8‬‬

‫‬‫‬‫‬‫‪-‬‬

‫‪4،2‬‬ ‫‪4،4‬‬ ‫‪4،6‬‬ ‫‪4،8‬‬

‫‪6،2‬‬‫‪6،4‬‬‫‪6،6‬‬‫‪6،8-‬‬

‫‪8،2‬‬‫‪8،4‬‬‫‪8،6‬‬‫‪8،8-‬‬

‫إذا أردﻧﺎ رﺳﻢ ﺗﻮزﻳﻊ ﻋﻴﻨﺔ‬ ‫اﻟﻤﺘﻮﺳﻄﺎت‪ ،‬ﻓﺈﻧﻨﺎ ﻧﻘﻮم‬ ‫ﺑﺤﺴﺎب ﻣﺘﻮﺳﻂ آﻞ‬ ‫ﻋﻴﻨﺔ أوﻻ آﻤﺎ هﻮ ﻣﻮﺿﺢ‬ ‫ﻓﻰ اﻟﺠﺪول‬

‫اﻟﻤﺘﻮﺳﻂ‬

‫ﺍﻟﺩﺭﺠﺎﺕ‬

‫ﺍﻟﺜﺎﻨﻴﺔ‬

‫ﺍﻷﻭﻟﻰ‬

‫ﺍﻟﻌﻴﻨﺔ‬ ‫) ‪(Χ‬‬

‫‪2‬‬

‫‪2‬‬

‫‪2‬‬

‫‪1‬‬

‫‪3‬‬

‫‪4‬‬

‫‪2‬‬

‫‪2‬‬

‫‪4‬‬

‫‪6‬‬

‫‪2‬‬

‫‪3‬‬

‫‪5‬‬

‫‪8‬‬

‫‪2‬‬

‫‪4‬‬

‫‪3‬‬

‫‪2‬‬

‫‪4‬‬

‫‪5‬‬

‫‪4‬‬

‫‪4‬‬

‫‪4‬‬

‫‪6‬‬

‫‪5‬‬

‫‪6‬‬

‫‪4‬‬

‫‪7‬‬

‫‪6‬‬

‫‪8‬‬

‫‪4‬‬

‫‪8‬‬

‫‪4‬‬

‫‪2‬‬

‫‪6‬‬

‫‪9‬‬

‫‪5‬‬

‫‪4‬‬

‫‪6‬‬

‫‪10‬‬

‫‪6‬‬

‫‪6‬‬

‫‪6‬‬

‫‪11‬‬

‫‪7‬‬

‫‪8‬‬

‫‪6‬‬

‫‪12‬‬

‫‪5‬‬

‫‪2‬‬

‫‪8‬‬

‫‪13‬‬

‫‪6‬‬

‫‪4‬‬

‫‪8‬‬

‫‪14‬‬

‫‪7‬‬

‫‪6‬‬

‫‪8‬‬

‫‪15‬‬

‫‪8‬‬

‫‪8‬‬

‫‪8‬‬

‫‪16‬‬

‫وﻳﻤﻜﻦ ﺗﻤﺜﻴﻞ ﺗﻮزﻳﻊ ﻋﻴﻨﺔ اﻟﻤﺘﻮﺳﻄﺎت‬ ‫ﺑﺎﺳﺘﺨﺪام اﻟﻤﺪرج اﻟﺘﻜﺮارى اﻟﺘﺎﻟﻰ ‪:‬‬ ‫اﻟﺘﻜﺮار‬

‫‪4‬‬ ‫‪3‬‬ ‫‪2‬‬ ‫‪1‬‬

‫‪9‬‬

‫‪8‬‬

‫‪7‬‬

‫‪6‬‬

‫‪5‬‬

‫‪4‬‬

‫‪3‬‬

‫‪2‬‬

‫‪1‬‬

‫‪0‬‬

The Theoretical Normal Curve

(from http://www.music.miami.edu/research/statistics/normalcurve/images/normalCurve1.gif

Properties of the Normal Curve: Theoretical construction Also called Bell Curve or Gaussian Curve Perfectly symmetrical normal distribution The mean of a distribution is the midpoint of the curve The tails of the curve are infinite Mean of the curve = median = mode The “area under the curve” is measured in standard deviations from the mean

Properties (cont.)

Has a mean = 0 and standard deviation = 1. General relationships: ±1 s = about 68.26% ±2 s = about 95.44% ±3 s = about 99.72%

68.26%

95.44% 99.72%

-5

-4

-3

-2

-1

0

1

2

3

4

5

The Normal Curve

„

„

„

34% of scores between the mean and 1 SD above or below the mean An additional 14% of scores between 1 and 2 SDs above or below the mean Thus, about 96% of all scores are within 2 SDs of the mean (34% + 34% + 14% + 14% = 96%)

Note: 34% and 14% figures can be useful to remember

Mean = 65 S=2 0.70 0.60 Relative Frequency

There are known percentages of scores above or below any given point on a normal curve

0.50 0.40

99.72% of cases 95.44% of cases 68.26% of cases

0.30 0.20 0.10 2% 14% 34% 34% 14% 2% 0.00 51 53 55 57 59 61 63 65 67 69 71 73 75 77 79 -3S -2S -1S 0 +1 S +2S +3S

The Standard Normal Curve “A standard score expresses a score’s position in relation to the mean of the distribution, using the standard deviation as the unit of measurement” “A z-score states the number of standard deviations by which the original score lines above or below the mean” Any score can be converted to a z-score.

X−X z= S