Lectures 1 & 2 PDF

# إحصاء رياضي - *step 1* محاضرة 241 - 2 ≤ 1 ## Chapters - Chapter 1: Frequency distributions and graphs - Chapter 2: Data description - Chapter 3: Moments - Chapter 4: Correlations and regressions equations - Chapter 5: Introduction to probability ## Chapter 1 ### Some definitions - **Statistics...

# إحصاء رياضي - *step 1* محاضرة 241 - 2 ≤ 1 ## Chapters - Chapter 1: Frequency distributions and graphs - Chapter 2: Data description - Chapter 3: Moments - Chapter 4: Correlations and regressions equations - Chapter 5: Introduction to probability ## Chapter 1 ### Some definitions - **Statistics science:** Statistics science is the branch of science which concerned with the scientific method of collecting, organizing, presenting, summarizing and analyzing statistical information (data) as well as drawing valid conclusion on the basis of such analysis. It could be simply defined as *science of data*. ### Divisions of Statistics - **Descriptive (وصفى)**: Collecting, organizing, presenting, summarizing and describing data - **Inferential (استدلالى)**: Analyzing: draw valid conclusion ### Important terms - **Population:** is the collection of all items or things under consideration - **Sample:** is a portion of the population selected for analysis ### Data - **Categorical (qualitative)**: ex: Eye color - **Numerical (quantitative)** - **Discrete (Counted item)**: Ex: number of children - **Continuous (Intervals or range)** ### Frequency distribution and frequency table - **Frequency distribution:** - **Category** | **F** - --- | --- - A | 3 - B | 5 - C | 2 - D | 10 - E | 17 - F | 8 - **Frequency table:** - **Interval** | **F** - --- | --- - 1-10 | 2 - 11-20 | 5 - 21-30 | 8 - 31-40 | 3 - 41-50 | 7 ### How to group the data? - How to form frequency table? - **① Range (R):** هو أكبر قيمة في البيانات مطروح منها أصغر قيمة في البيانات - R = max - min - **② No of classes (K):** عدد البيانات =K = n ( لا بد أن يكون عرو صحيح) - No of data - **③ Class size (W):** ه نحسب حجم الفترات - W = R/K (لابد أن يكون عدد صحيح) - Class size ### Example 1: The following are the marks of 50 students in statistic: - 48, 70, 60, 47, 51, 55, 59, 63, 68, 63 - 47, 53, 72, 53, 67, 62, 64, 70, 57, 56 - 48, 51, 58, 63, 65, 62, 49, 64, 53, 59 - 63, 50, 61, 67, 72, 56, 64, 66, 49, 52 - 62, 71, 58, 53, 63, 69, 59, 64, 73, 56 #### Questions: - (a) Construct a frequency table for the above data - (b) How many students scored between 51 and 63? - (c) How many students scored above 51? #### Solution 1 - (a) **Range** - R = 73 - 47 = 26 - **Number of classes K** - K = √50 = 7.07 ~ 7 - **Class size W** - W = 26/7 = 3.7 ~ 4 #### Frequency Table - **Intervals** | **Tally** | **Frequency (F)** - --- | --- | --- - 47-51 | | 7 - 51-55 | | 7 - 55-59 | | 7 - 59-63 | | 8 - 63-67 | | 11 - 67-71 | | 6 - 71-75 | | 4 - Σ = 50 **Table (1)** - (b) 7 + 7 + 8 = 22 - (c) 7 + 7 + 8 + 11 + 6 + 4 = 43 - or 50 - 7 = 43 ### Glass Mark (Mid-Point) - For an interval [a-b] ⇒ Mid-Point = a+b/2 ### Histogram, polygon, curve (smooth curve) and pie chart #### Example 2 Given Table (1) in Example 1, draw Histogram, Polygon and the curve of distribution. #### Solution: - **Histogram** - A bar graph showing the frequency distribution of a data set. The height of each bar represents the frequency of a particular interval. - **Polygon** - A line graph that connects the midpoints of the tops of the bars in a histogram. The x-axis and y-axis are the same as the histogram, but the y-axis represents the frequencies. - **Curve** - A smoother version of the polygon. #### Example 3: In ADAS international school, the lesson periods for each week are given as: - English 7, Maths 10 , Biology 3, Physics 4, Chemistry 3, others 9. - Draw a pie chart to illustrate this information. #### Solutions: **Angle of sector** - **Subject** | **No.** | **Angle of sector** - --- | --- | --- - English | 7 | 7/36 * 360 = 70° - Maths | 10 | 10/36 * 360 = 100° - Biology | 3 | 3/36 * 360 = 30° - Physics | 4 | 4/36 * 360 = 40° - Chemistry | 3 | 30° - others | 9 | 90° ### Upper cumulative frequency (U.G.F or U-Ogives) & Lower cumulative frequency (L.G.F or L-Ogives): #### Example 4: The following data represent the record high temperature for each of the 50 U.S. States. - 112, 100, 127, 120, 133, 118, 105, 110 , 109, 112 - 110, 118, 117, 116, 118, 122, 114, 114, 105, 109 - 107, 112, 114, 115, 118, 117, 118, 122, 106, 110 - 116, 108, 110, 121, 113, 120, 119, 111, 104, 111 - 120, 113, 120, 117, 105, 110, 118, 112, 114, 114. #### Questions - (a) Construct the frequency table, showing on it class mark. - (b) Draw histogram and curve for this data. - (c) Construct U.C.F & L.C.F and draw its then find the median. - (d) How many states record a high temperature less than 113. #### Solutions: - (a) **Range** - R = 133 - 100 = 33 - **Number of classes K** - K = √50 = 7.07~7, - **Class size W** - W = 33/7 = 4.71 ~ 5 #### Frequency Table - **Intervals** | **Class Mark** | **Tally** | **Frequency** - --- | --- | --- | --- - 100 - 105 | 102 ½ | // | 2 - 105 - 110 | 107 ½ | //// | 8 - 110 - 115 | 112½ | ///// ///// /// | 19 - 115 - 120 | 117 ½ | ///// /// | 13 - 120 - 125 | 122 ½ | ///// | 6 - 125 - 130 | 127 ½ | // | 1 - 130 - 135 | 132 ½ | | 1 - Σ = 50 - (b) **Histogram** - Similar description to previous histogram. - **Curve** - Similar description to previous curve. - (c) **U.G.F** - **Intervals** | **f** - --- | --- - Less than 100 | 0 - // 105 | 2 - // 110 | 10 - // 115 | 29 - // 120 | 42 - // 125 | 48 - // 130 | 49 - // 135 | 50 - **L.C.F** - **Intervals** | **f** - --- | --- - 100 and more | 50 - 105 // | 48 - 110 // | 40 - 115 // | 21 - 120 // | 8 - 125 // | 2 - 130 // | 1 - 135 // | 0 - (d) 22 states #### Example 5: Given the following frequency table: - **Intervals** | **F** - --- | --- - 50-60 | 8 - 60-70 | 10 - 70-80 | 16 - 80-90 | 15 - 90-100 | 10 - 100-110 | 8 - 110-120 | 3 Construct: U-ogives & L-Ogives tables. #### Solutions - **V. C-F** - **Intervals** | **f** - --- | --- - Less than 50 | 0 - // 60 | 8 - // 70 | 18 - // 80 | 34 - // 90 | 49 - // 100 | 59 - // 110 | 67 - // 120 | 70 - **L.G.F** - **Intervals** | **f** - --- | --- - 50 and more | 70 - 60 // | 62 - 70 // | 52 - 80 // | 36 - 90 // | 21 - 100 // | 11 - 110 // | 3 - 120 // | 0 - **H.W 1:** In Example 5, draw Histogram and find the median graphically.