Statistics for Play Therapists I

Statistics for Play Therapists IDebi Maskell-GrahamMA Practice-based Play Therapy

OverviewWe are able to collect and analyse data in many ways for research purposes.Two types of statistics; descriptive and inferential.Descriptive Statistics consists of methods for organizing, displaying and describing data by using tables, graphs, and summary measures (describing things).Inferential Statistics is a process of describing the population based on the sample results (making inferences).The focus of this webinar is descriptive statistics.

Typically, when we look at data we are interested in the measures of central tendency of the data:mean (the average)median (the middle score) mode (the most popular score)

and the indicators for the spread of the data:the range interquartile rangestandard deviation (data reliability)

OutcomesThis first section explains these quantities - the mean, median, mode, range and interquartile range - for a number of research situations for which data has been collected.

By the end of the webinar students will;Understand the potential value and use of statistics in research projects.be able to define basic terms used in statistics.be competent to compute simple measures of central tendency.be able to construct frequency tables, histograms and frequency polygons that display these measures of central tendency.

Example datasetAn MA student wanted to know how many siblings each of her study children had.

There were 6 families in total and the results were:2, 0, 3, 2, 1, 2

MeanThe mean is the average number of siblings per study child and is calculated by adding all the sibling numbers up and dividing by the number of study children:

2+0+3+2+1+2 = 1.83 (you can instantly see a problem with just stating numbers!!)

Exercises: calculating the mean1.) Five employees of a Department Store earned the following hourly wages: 4.35, 3.67, 3.36, 5.00 and 4.82. Find the average hourly rate of pay.2.) During a six week period Mary worked the following number of hours per week: 40, 42 , 37 , 48, 45, 44 . Find the average number of hours that Mary worked per week.3.) Eight cars were priced at 10,499; 11,988; 7,444; 5,995; 14,999; 6,492; 10,750; and 7,937. What is the arithmetic mean of the prices?4.) Last winter a homeowner purchased 504 gallons of heating oil at an average cost of 1.23 per gallon. If he paid 1.19 per gallon for the first 354 gallons, what was the total cost of the remaining oil purchased?5.) Louise allows herself an average of 2.50 a day for lunch at work. If she spent 2.00 on Monday, 2.75 on Tuesday, 2.25 on Wednesday, and 2.50 on Thursday, how much may she spend for lunch on Friday?6.) Jemma purchased ten dozen bread rolls as follows: 2 dozen 1.19, 1 dozen 1.88, 3 dozen 1.94, and 4 dozen 1.28. What average price per dozen did she pay for the ten dozen rolls?7.) On holiday, Jack bought petrol as follows: 10 litres at 1.29 per litre, 15 litres at 1.19, and 12 litres at 1.23. What was the average price per litre?

ModeThe mode is the most common or frequent number of siblings per study child.0,1,2,2,2,3 - the most frequent number is 2 siblings

Exercises: calculating the mode1.) Find the mode of the given data.

a.) 3, 5, 6, 7, 6, 2.b.) 14, 15, 14, 17, 15, 15, 18, 19.c.) 231, 237, 248, 244.d.) 84, 86, 87, 84, 86, 89, 90, 87, 87, 84, 86.e.) 29, 30, 31, 28, 29, 35.

MedianThe median is the middle number of the range of numbers = 2 siblings

0, 1, 2, 2, 2, 3

Exercises1.) Find the median of the given data:a.) 18.2, 16.8, 13.3, 19.4, 17.6b.) 68, 62, 64, 60, 63, 61, 59c.) 237, 225, 230, 228, 236, 232d.) 70, 74, 71, 72, 80, 78, 75, 69e.) 18.42, 16.74, 19.88, 15.64, 24.38, 14.76

2.) Write mean or median to complete the sentence for the following data.a.) 5, 6, 2, 9, 8. The -------- is 6.b.) 11, 2, 3, 4, 6, 10. The ---------- is 6.c.) 9.7, 4.2, 6.3, 8.5, 7.4. The ----------- is 7.4.d.) 85, 83, 86, 90, 88, 91, 93. The ------------- is 88.e.) 3.60, 2.85, 4.90, 4.50, 5.00, 2.75. The ----------- is 4.05.

3.) Are the following statements true or false?a.) The median of 12, 5, 6, 13, 10, 9, 7, 8, 14 is 10.b.) The median of 6.5, 4.0, 8.5, 2.5, 5.0, 3.5 is 4.5.c.) The mean of 2.8, 7.6, 5.4, 8.2, 3.9 is 5.4.d.) The mean of 12.6, 6.8, 5.9, 10.7 is 9.e.) The median of 184, 200, 150, 148, 178 is 178.

Comparing mean, median and modeFor one group of results collected from a survey the mean might be near both the median and the mode. In this case it can be said that the distribution is a balanced distribution, with all three measures of the centre of the distribution being close to each other.For a particular group of results the value of the mean might not be near the median and the mode. In this situation it can be interpreted that the spread of the data is not balanced about the middle of the distribution. This is due to extreme values at one end of the distribution. This distribution is called a skewed distribution.

Range and interquartile rangeOne indicator of the spread of data is given by the range - the gap between the lowest and the highest values. However, the range CAN include values which are uncharacteristically high or low in your sample. A more accurate way is to present the interquartile range of your data excluding the data which can skew the overall range.

RangeThe average age of US women getting married over the last fifty years can becollected from census data.

Census yearWomens age194723195422.6196121.8196621.5197121.4197622.2198123.3198624.9199126199627.2

The table shows that the highest age at which women married over the past 50years was 27.2 years during 1986. The lowest age which women married at was21.4 years during 1971. It can be said therefore that the range between thelowest and the highest average age was approximately 6 years - 5.8 years to beexact.

Exercises2.) Find the range of the given data.a.) 3.6, 9.2, 5.8, 7.4, 12.1.b.)22.54, 19.82, 50.00, 35.60, 42.78, 15.63.c.) 70 64, 98, 69, 82, 85, 59.d.) 12, 6 , 13, 8 , 11, 15 .e.) 26.3, 9.27, 15.7, 28.9, 18.8.

3.) Write mean, mode, median or range to complete the sentence for the given data.a.) 17, 19, 17, 15, 20, 16, 18. The ----- is 17.b.) 78, 68, 82, 96, 84, 90, 76. The ------ is 82.c.) 42, 69, 53, 75, 97, 88, 38. The -------- is 69.d.) 22, 10, 42, 39, 27, 32, 49. The ----------is 39.

Interquartile rangeWe know that the median divides the data into two halves. We also know that for a set of n ordered numbers, the median is the (n+1)/2 th value.Similarly, the lower quartile divides the bottom half of the data into two halves and the upper quartile divides the upper half of the data into two halves.Lower quartile is the (n+1)/4 th value.Upper quartile is the 3(n+1)/4 th value.

Example1.) Find the median, lower quartile and upper quartile forthe following data: 11, 4, 6, 8, 3, 10, 8, 10, 4, 12, 31Answer:Ordering the data, we get 3, 4, 4, 6, 8, 8,10, 10, 11, 12, 31There are 11 numbers.The median is the (11+1)/2 th = 6th value.The lower quartile is the (11+1)/4 th = 3rd value.The upper quartile is the 3(11+1)/4 th = 9th value.Therefore the median is 8, the lower quartile is 4 and theupper quartile is 11.

3, 4, 4, 6, 8, 8, 10, 10, 11, 12, 31

The interquartile range is the difference between the upperquartile and the lower quartile. In this example, theinterquartile range is 11 - 4 = 7.

Considering outliers2.) The interquartile range ignores extreme values(calls them outliers).

The range includes extreme values.

Look at this set of data:-1, 5, 7, 8, 9, 12, 13, 15, 17, 18, 35,The interquartile range is 17 - 7 = 10,The range is 35 - 1 = 34.In cases such as these it is often preferable to usethe interquartile range when comparing the data.

Frequency tableLarge amounts of information can easily be organized, read and understood by listing data in a frequency table. A frequency table is a chart in which the members of a set are tallied, and the total count for each item is recorded. If the set has a wide range of elements, it may be divided into equal intervals to make the frequency table shorter.

Finding the mean using a frequency tableSara wanted to know the ages (in whole years) of young people who had accessed play therapy over a year. She conducted a survey and her results are shown below:131411121215131412161511111211121416141514141312131111141213

To find the mean add all the ages together andDivide by the total number of children.

Using a frequency tableIf you type all those ages into a calculator it is highly likely that youwould make a mistake or forget where you were up to.It would be better if you could see these results displayed in afrequency table:

AgeFrequency116127135147153162

The frequency table shows us that there are six children aged 11, seven childrenaged 12, five children aged 13...etc.To find the sum of their ages, calculate:(6 11) + (7 12) + (5 13) + (7 14) + (3 15) + (2 16) = 390The total number of children is 6 + 7 + 5 + 7 + 3 + 2 = 30So the mean age is 390 30 = 13

ExampleThe prices of 25 different television sets arelisted below. Show the data in a frequencytable. Determine the median price and therange of prices.Prices of flat screen TVs

219399400359318360200480247430475250260278397499480427387435314425450287498

Calculating the medianand rangeSolution: Establish six intervals of 50 each. Tally the number of items in eachinterval.Price rangeTallyFrequency

200-249III3250-299IIII4300-349II2350-399IIIII5400-499IIIII5450-500IIIIII6The median may be determined as follows:(25 + 1) divided by 2 = 26 divided by 2 = 13The l3th item of the set is the median price. The l3th item is the fourth item inthe interval 350 399. The items in this interval are: 359, 360, 387, 397, 399. Themedian price is 397.

The range may be determined as follows:Range = 498 minus 200 = 298.

ExercisesExercises: Solve the following problems.1.) The high temperatures in twenty-five cities on June 29were as follows:

6964618787587158877270528210711273687887687173909683

a.) Arrange the temperatures in intervals and make aFrequency table for the set of data.b.) What is the range of the temperatures?c.) What is the mode of the temperatures?d.) What is the median temperature?e.) What is the mean temperature?

ExerciseThe salaries of thirty people are listed below.

12,50023,90018,75024,00014,00018,75011,57025,0009,20015,00024,00022,00020,50012,50017,30010,98015,55018,75018,00016,20032,00013,00022,00035,00021,000

a.) Arrange the salaries in intervals and make a frequency table forthe set of data.b.) What is the range of the salaries?c.) What is the median salary?d.) What is the mean salary?

HistogramsThe frequency of a set of numbers can be represented graphically on a histogram. A histogram is a bar graph on which the bars are adjacent to each other with no space between them. To construct a histogram, arrange the data in equal intervals. Represent each interval on the horizontal axis of the graph. Represent the frequency of items in the interval on the vertical axis of the graph.

Example: creating a histogramCreating a histogram displaying the top ten women's figureskating scores for the 2010 Winter Olympics

Figure SkatingYu-Na Kim228.56Mao Asada205.5Joanie Rochette202.64Mirai Nagasu190.15Miki Ando188.86Laura Lepisto187.97Rahail Flatt182.49Akiko Suzuki181.44Alena Leonova172.46Ksenia Makarova171.91

The x-axis needs to span from at least 171 to 229 in order toaccommodate all of the data.

StagesSince we are not given a specific bin width we can create a histogram with whatever width we choose. It is easiest to pick a nice round number like 5 or 10, but it really depends on how the data is spread. We choose a bin width of 10 points.Now we look at the data and find how many skaters scored in each interval. The highest will be 180-190, with a frequency of 4, so we need to make sure that our y-axis spans to at least 4.Finally we plot the frequency of each interval: 2 between 170-180, 4 between 180-190, 1 between 190-200, 2 between 200-210, and 1 between 220-230.

Final histogram

Exercise: So many things you can do with the data!!A test was scored on the basis of 1 to 20 points. The scores obtained by thirtystudents are as follows: 20, 12, 18, 11, 20, 8, 1, 5, 10, 12, 15, 15, 18, 13, 19,20, 18, 15, 16, 18, 17, 14, 20, 19, 13, 16, 12, 18, 20, 8.

1.) Arrange the scores in intervals and construct a histogram depicting the number of students who scored in each interval. 2.) In which interval do most test scores lie? 3.) Which interval contains the least number of test scores? 4.) What is the average number of points scored? 5.) What is the range of test scores? 6.) What is the median test score? 7.) What percent of all scores lie in an interval from eleven to fifteen inclusive? 8.) What is the mode of the test scores? 9.) How many students received a grade of 75% or better?10.) If 60% of the total possible points represents a passing grade, how many students passed the test?11.) How many more students scored in the interval 16 to 20 than in the interval 11 to 15?12.) Did any student receive the median score?13.) How many students scored higher than the mean?14.) How many students scored below the median?15.) What percent of the students failed the test?

Frequency polygonA frequency polygon is a line graph which can be used to represent the frequency of a set of numbers. It is formed by connecting a series of points. The abscissa (horizontal) of each point is the midpoint of the interval in which the point lies.The ordinate (vertical) of each point is the frequency for the interval. The polygon is closed at each end by drawing a line from the endpoints to the horizontal axis at the midpoint of the next interval.

ExampleA frequency polygon gives the idea about the shape of the data distribution.The two end points of a frequency polygon always lie on the x-axis.The frequency polygon shown above represents the number of vehicles that passes through a particular route in different hours. To draw the above diagram first a histogram is drawn and then a line graph is drawn through the midpoints of the top of the bars.

ExerciseThe students in a certain class received thefollowing marks on a test:

949672908784729060689378688094758790877881857087

1.) Group the data in intervals and construct afrequency polygon to show the number of studentsin each interval.2.) What is the mode?3.) What is the mean score?4.) What is the range?

ReferencesAdditional content for this webinar wasadapted from:

http://www.yale.edu/ynhti/curriculum/unitshttp://www.dummies.com/how-to/content/statistics-for-dummies-cheat-sheet.html

Books:

Diamond, I. and Jeffries, J. (2001) Beginning withStatistics. Sage Publications Ltd: London.