Key Formula for Research Method IRC010 Part 1: Chapter 1 ~7
Made by Minjeong Kim
2024475001
Chapter 2 Visual Description of Data:
1) Sturges Rule: Sturges' rule is a formula used to determine the number of bins, or classes,
to use when creating a histogram for a dataset:
Formula: k=1+3.322 x 𝐥𝐥𝐥𝐥𝐥𝐥 𝟏𝟏𝟏𝟏 𝒏𝒏
k = number of classes (bins), n = number of observations in the dataset, log n = logarithm
to base 10
Example Question: For 𝟐𝟐𝟑𝟑 =8, so 3= 𝐥𝐥𝐥𝐥𝐥𝐥 𝟐𝟐 𝟖𝟖
Chapter 3 Statistical Description of Data:
1) Measures of Central tendency (the center)
a. Mean, population mean=𝝁𝝁, sample mean=x̄
𝑺𝑺𝑺𝑺𝑺𝑺 𝒐𝒐𝒐𝒐 𝒂𝒂𝒂𝒂𝒂𝒂 𝒗𝒗𝒗𝒗𝒗𝒗𝒗𝒗𝒗𝒗𝒗𝒗
𝒕𝒕𝒕𝒕𝒕𝒕 𝒏𝒏𝒏𝒏𝒏𝒏𝒏𝒏𝒏𝒏𝒏𝒏 𝒐𝒐𝒐𝒐 𝒗𝒗𝒗𝒗𝒗𝒗𝒗𝒗𝒗𝒗𝒗𝒗
, ∑ 𝒙𝒙𝒊𝒊
Population mean: 𝝁𝝁=
𝑵𝑵
∑ 𝒙𝒙𝒊𝒊
Sample mean: x̄=
𝒏𝒏
b. Weighted mean: (When you have grouped data)
∑ 𝒙𝒙𝒊𝒊 𝒘𝒘𝒊𝒊
𝝁𝝁 = ∑ 𝒘𝒘𝒊𝒊
c. Median: 1) Put the data in array 2A. IF the data set has odd numbers -> median=middle
value 2B. Data set has even number of numbers -> median=average of two middle
values
d. Mode: the most frequent value
Just one mean/median for given data set. There may be more than one value for the mode.
-If mean=median=mode: the shape of the distribution is symmetric.
-If mode<median<mean: the shape of the distribution trails to the right. (Positively skewed)
-If mean < median<mode: the shape of the distribution trails to the left. (Negatively skewed).
2) The Spread
a. Range: Largest value minus smallest value
,b. Residuals: Difference between each data value in the set and the group mean
Population: 𝒙𝒙𝒊𝒊 − 𝝁𝝁
Sample: 𝒙𝒙𝒊𝒊 − x̄
c. MAD (Mean Absolute Deviations): Summing the absolute values of all residuals and
dividing by the number of values in the set
(∑|𝒙𝒙𝒊𝒊 −𝝁𝝁|)
Population MAD:
𝑵𝑵
(∑|𝒙𝒙𝒊𝒊 −𝐱𝐱𝐱|)
Sample MAD:
𝒏𝒏
d. Variance (The Spread)
∑(𝒙𝒙𝒊𝒊 −𝝁𝝁)𝟐𝟐
Population 𝝈𝝈𝟐𝟐 =
𝑵𝑵
∑(𝒙𝒙𝒊𝒊 −𝐱𝐱𝐱)𝟐𝟐
Sample 𝒔𝒔 = 𝟐𝟐
𝒏𝒏−𝟏𝟏
e. Standard Deviation
Population: √𝝈𝝈𝟐𝟐 Sample: √𝒔𝒔𝟐𝟐
Coefficient of Variation (CV)
𝝈𝝈
CV= x 100% (relative amount of dispersion in the data)
𝝁𝝁
, 3) Relative Position-Quartiles
a) Quartiles
Most used quantiles=quartiles
Quartiles -> divide the values of data set into 4 subsets of equal size. Each account for
25%.
1) Arrange the N data values from smallest to largest.
𝑵𝑵+𝟏𝟏
2) First Quartile 𝑸𝑸𝟏𝟏 =data value at position
𝟒𝟒
𝟐𝟐(𝑵𝑵+𝟏𝟏)
3) Second Quartile 𝑸𝑸𝟐𝟐 = data values at position
𝟒𝟒
𝟑𝟑(𝑵𝑵+𝟏𝟏)
4) Third Quartiles 𝑸𝑸𝟑𝟑 = data values at position
𝟒𝟒
4) What is standardized value?
How far above or below the individual value is compared to the population mean in units of
standard deviation
Made by Minjeong Kim
2024475001
Chapter 2 Visual Description of Data:
1) Sturges Rule: Sturges' rule is a formula used to determine the number of bins, or classes,
to use when creating a histogram for a dataset:
Formula: k=1+3.322 x 𝐥𝐥𝐥𝐥𝐥𝐥 𝟏𝟏𝟏𝟏 𝒏𝒏
k = number of classes (bins), n = number of observations in the dataset, log n = logarithm
to base 10
Example Question: For 𝟐𝟐𝟑𝟑 =8, so 3= 𝐥𝐥𝐥𝐥𝐥𝐥 𝟐𝟐 𝟖𝟖
Chapter 3 Statistical Description of Data:
1) Measures of Central tendency (the center)
a. Mean, population mean=𝝁𝝁, sample mean=x̄
𝑺𝑺𝑺𝑺𝑺𝑺 𝒐𝒐𝒐𝒐 𝒂𝒂𝒂𝒂𝒂𝒂 𝒗𝒗𝒗𝒗𝒗𝒗𝒗𝒗𝒗𝒗𝒗𝒗
𝒕𝒕𝒕𝒕𝒕𝒕 𝒏𝒏𝒏𝒏𝒏𝒏𝒏𝒏𝒏𝒏𝒏𝒏 𝒐𝒐𝒐𝒐 𝒗𝒗𝒗𝒗𝒗𝒗𝒗𝒗𝒗𝒗𝒗𝒗
, ∑ 𝒙𝒙𝒊𝒊
Population mean: 𝝁𝝁=
𝑵𝑵
∑ 𝒙𝒙𝒊𝒊
Sample mean: x̄=
𝒏𝒏
b. Weighted mean: (When you have grouped data)
∑ 𝒙𝒙𝒊𝒊 𝒘𝒘𝒊𝒊
𝝁𝝁 = ∑ 𝒘𝒘𝒊𝒊
c. Median: 1) Put the data in array 2A. IF the data set has odd numbers -> median=middle
value 2B. Data set has even number of numbers -> median=average of two middle
values
d. Mode: the most frequent value
Just one mean/median for given data set. There may be more than one value for the mode.
-If mean=median=mode: the shape of the distribution is symmetric.
-If mode<median<mean: the shape of the distribution trails to the right. (Positively skewed)
-If mean < median<mode: the shape of the distribution trails to the left. (Negatively skewed).
2) The Spread
a. Range: Largest value minus smallest value
,b. Residuals: Difference between each data value in the set and the group mean
Population: 𝒙𝒙𝒊𝒊 − 𝝁𝝁
Sample: 𝒙𝒙𝒊𝒊 − x̄
c. MAD (Mean Absolute Deviations): Summing the absolute values of all residuals and
dividing by the number of values in the set
(∑|𝒙𝒙𝒊𝒊 −𝝁𝝁|)
Population MAD:
𝑵𝑵
(∑|𝒙𝒙𝒊𝒊 −𝐱𝐱𝐱|)
Sample MAD:
𝒏𝒏
d. Variance (The Spread)
∑(𝒙𝒙𝒊𝒊 −𝝁𝝁)𝟐𝟐
Population 𝝈𝝈𝟐𝟐 =
𝑵𝑵
∑(𝒙𝒙𝒊𝒊 −𝐱𝐱𝐱)𝟐𝟐
Sample 𝒔𝒔 = 𝟐𝟐
𝒏𝒏−𝟏𝟏
e. Standard Deviation
Population: √𝝈𝝈𝟐𝟐 Sample: √𝒔𝒔𝟐𝟐
Coefficient of Variation (CV)
𝝈𝝈
CV= x 100% (relative amount of dispersion in the data)
𝝁𝝁
, 3) Relative Position-Quartiles
a) Quartiles
Most used quantiles=quartiles
Quartiles -> divide the values of data set into 4 subsets of equal size. Each account for
25%.
1) Arrange the N data values from smallest to largest.
𝑵𝑵+𝟏𝟏
2) First Quartile 𝑸𝑸𝟏𝟏 =data value at position
𝟒𝟒
𝟐𝟐(𝑵𝑵+𝟏𝟏)
3) Second Quartile 𝑸𝑸𝟐𝟐 = data values at position
𝟒𝟒
𝟑𝟑(𝑵𝑵+𝟏𝟏)
4) Third Quartiles 𝑸𝑸𝟑𝟑 = data values at position
𝟒𝟒
4) What is standardized value?
How far above or below the individual value is compared to the population mean in units of
standard deviation
